This answer is very long.
SHORT ANSWER
To make it short, your emperor has to do a lot of military conquering to gain political power over larger and larger areas. Once he has sufficient political and military might, there would be no chance of any type of a political challenge by a pope.
LONG ANSWER
Your question is based on a flawed premise, equating political power and authority with authority in a religious institution. Medieval and later antipopes claimed to have political power on the basis of being the self proclaimed successors of St. Peter.
Christendom is not a state. It is the region where some of the people, perhaps a tiny minority, in every state are Christians, and today after centuries of world wide missionary efforts by Christians probably includes every state on the planet Earth.
Or in a more narrow sense it is the region where most of the people in every state are Christians to some degree. At the present time Christendom includes almost all the states in the continents of North America, South America, Europe, and Australia, and many other regions scattered through Africa, Asia, & various islands.
The only way for a ruler to become the ruler of Christendom is to either exterminate all Christians outside his own present realm, thus reducing Christendom to his own present realm, or else add all the tens or hundreds of states in Christendom to his own state by conquest and/or negotiation.
In the vague Late Antique/Early Medieval time frame of your question Christendom would include the region around the Mediterranean Sea, with extensions deep into Europe, Africa, and Asia. How far it penetrated into those continents would depend on whether Christendom was limited to lands with majority Christian populations or if lands with minority Christian populations were also included.
If you define Christendom in that era as including lands and states with minority Christian populations then Christendom will extend far into Asia, and possibly the majority of all Christians will be living scattered around Asia in states ruled by Hindus, Buddhists, Shamanists, Zoroastrians, and Muslims depending on the time and place.
And of course the vast majority of non Christian rulers of lands containing Christian minorities wouldn't and don't see any reason to give up any of their power by causing all of Christendom to be united. And thus if one's goal is to unite all Christendom in the broad sense many realms with minority Christian populations would and will have to be conquered in war and annexed.
So perhaps your question only applies to an attempt to unite all of Christendom in the narrow sense, restricted to land and states where Christians are in the majority and control the government more or less completely.
According to Mathew 22:15-22 (New Revised Standard Version translation):
15 Then the Pharisees went and plotted to entrap him in what he said.
16 So they sent their disciples to him, along with the Herodians, saying, “Teacher, we know that you are sincere, and teach the way of God in accordance with truth, and show deference to no one; for you do not regard people with partiality.
17 Tell us, then, what you think. Is it lawful to pay taxes to the emperor, or not?”
18 But Jesus, aware of their malice, said, “Why are you putting me to the test, you hypocrites? 19 Show me the coin used for the tax.” And they brought him a denarius.
20 Then he said to them, “Whose head is this, and whose title?”
21 They answered, “The emperor’s.” Then he said to them, “Give therefore to the emperor the things that are the emperor’s, and to God the things that are God’s.”
22 When they heard this, they were amazed; and they left him and went away.
https://www.biblegateway.com/passage/?search=Matthew+22%3A15%E2%80%9322&version=NRSV[1]
According to Mark 12:13-17 (New International Version translation):
13 Later they sent some of the Pharisees and Herodians to Jesus to catch him in his words.
14 They came to him and said, “Teacher, we know that you are a man of integrity. You aren’t swayed by others, because you pay no attention to who they are; but you teach the way of God in accordance with the truth. Is it right to pay the imperial tax[a] to Caesar or not?
15 Should we pay or shouldn’t we?”
But Jesus knew their hypocrisy. “Why are you trying to trap me?” he asked. “Bring me a denarius and let me look at it.”
16 They brought the coin, and he asked them, “Whose image is this? And whose inscription?”>
“Caesar’s,” they replied.
17 Then Jesus said to them, “Give back to Caesar what is Caesar’s and to God what is God’s.”
And they were amazed at him.
https://www.biblegateway.com/passage/?search=Mark+12%3A13-17&version=NIV2
Luke 20:20-26 (King James Version Translation):
20 And they watched him, and sent forth spies, which should feign themselves just men, that they might take hold of his words, that so they might deliver him unto the power and authority of the governor.
21 And they asked him, saying, Master, we know that thou sayest and teachest rightly, neither acceptest thou the person of any, but teachest the way of God truly:
22 Is it lawful for us to give tribute unto Caesar, or no?
23 But he perceived their craftiness, and said unto them, Why tempt ye me?
24 Shew me a penny. Whose image and superscription hath it? They answered and said, Caesar's.
25 And he said unto them, Render therefore unto Caesar the things which be Caesar's, and unto God the things which be God's.
26 And they could not take hold of his words before the people: and they marvelled at his answer, and held their peace.
https://www.biblegateway.com/passage/?search=Luke+20%3A20-26&version=KJV3
The only logical interpretation of those statements is that all Christians must worship only the Christian God and no other gods - otherwise they would be giving to other gods that which is God's - and must be subject to the political and military authority of the Roman Emperor and pay taxes to the Roman Emperor and no other ruler - otherwise they would be giving to other rulers that which was the Roman Emperor's.
And since Christians were also commanded to convert everyone in the world to Christianity, eventually every christian would be a subject of the Roman Emperor and everyone in the world would be a Christian and so a subject of the Roman Emperor, and so wars would become impossible since there would no independent states left to fight wars.
Different interpretations of the saying would be implying that the two phrases it was composed of were different in meaning and intention even though identical in form, and would imply that Jesus Christ was like some Human speaker in his inability to use divine knowledge and divine intelligence to say exactly what he meant.
Furthermore, interpreting that saying as anything except a command for all Christians everywhere and in every time to be part of the Roman Empire and have no other government except for the Roman Empire is implying that God doesn't care about establishing peace or preventing war.
Obviously if all Christians had always been loyal to the Roman Empire and only the Roman Empire it would have saved a hundred million people from being slaughtered in wars between various independent Christian states in the 20th century alone.
Think of all the death and destruction, horror and suffering, which will result in wars fought between 2019 and 2119, or between 2019 and 3019, or between 2019 and 1002019, or in all the future of the Human race and its possible descendants, if all of mankind is not united under one government to prevent wars.
So clearly it would be an insult to the God of the Christians to claim that he was not good enough to want to prevent future wars and so did nothing to prevent future wars by forbidding Christians from being subjects of any ruler except for the Roman Empire.
And of course for many centuries Christian theologians scholars living in various independent states not part of the Roman Empire have been interpreting this phrase wrongly because they don't dare to realize that they are guilty of disobeying a commandment of Jesus Christ.
So your hypothetical fictional Roman Emperor can proclaim that all Christians everywhere have to be subjects of the Roman Empire or else be equally as un Christian as if they worshiped other gods instead of the Christian God. But it probably wouldn't do him much good sending missionaries to other Christian lands to spread that message. He would probably have to conquer those other Christian lands one by one with his armed forces and then decree that any bishop or priest who doesn't preach that interpretation of the Gospels is a heretic, in order to successfully spread that interpretation among the Christians in those lands.
I note that in real history Constantinople replaced Rome as the main center of political power in Christendom about as soon as there was a Christendom. In the 2nd half of the 3rd century emperors ceased to reside in Rome and lived in and ruled from several other cities. Then in 330 Constantinople was consecrated as a new capital city and quickly became more important than Rome, since it was an imperial residence and administrative capital for generation after generation and century after century, while the eastern part of the Roman Empire flourished and the western part rapidly was taken over by barbarians, and was not ruled from Rome anyway.
Once the final division of the Roman Empire into eastern and western zones of rule with their separate emperors happened in 395, the eastern portion, ruled from Constantinople, was clearly more powerful than the western section ruled from Milan or Ravenna. And it remained the most powerful realm in the Mediterranean region and/or western Eurasia on and off for centuries.
And since those states in the region which from time to time were more powerful than the eastern Roman Empire were usually ruled by non Christians and/or had non Christian majorities, the eastern part of the Roman Empire remained the most powerful state in Christendom for centuries longer than it remained the most powerful state in western Eurasia and the Mediterranean.
Thus Constantinople remained the greatest power center in Christendom, with only brief interruptions, until perhaps about 1180 when Emperor Manuel I died or maybe about 1203/04 with the capture of Constantinople by the 4th crusade.
So your fictional emperor basically doesn't have to do anything to make himself the most powerful ruler in Christendom, since an eastern Roman emperor ruling in most eras of late Antiquity/Early Middle Ages would already be the most powerful ruler in Christendom.
But if your fictional emperor wants to make himself the supreme ruler of all Christendom he does have work to do since Christian missionaries began converting people outside of the Roman Empire early in the history of Christianity, and various tribes and countries converted wholesale to Christianity about the same time as the Roman Empire did.
So your Late Antique/Early Medieval emperor will have to do a lot of conquering with his armed forces if he wants to rule all of Christianity.
First he would have to reconquer all of the western part of the Roman Empire that had ever been conquered by barbarians or revolted against the Empire. And then he would have to conquer barbarian kingdoms beyond the former borders of the Empire, in the British Isles and Germany, which had been converted to Christianity, or at least force their rulers to become his obedient clients, tributaries, or vassals.
And then he would have to do some expanding in the east, in Europe, Asia, and Africa, to conquer and annex various Christian majority countries beyond the imperial borders, or at least force their kings, great kings, or kings of kings to become his obedient vassals.
And if your fictional Late Antique/Early Medieval emperor does all that conquering, annexing, and vassalizing, he will rule a much larger area that even the most powerful emperors like Trajan (Ad 98-117) or Septimius Severus (AD 193-211) did.
Thus there will be no chance of any sort of political challenge by any pope.
But what about a religious challenge over doctrine by some bishop, archbishop, and Patriarch of Rome?
Well, the emperor can remind everyone that the Gospels are unclear whether the authority given by Christ to St. Peter was supposed to be personal to Peter and die with him or else permanent and passed on to Peter's successor. And if Peter's authority over the Church was supposed to be permanent and passed on to his successor, then who was Peter's rightful successor?
Peter became the leader of the Christian community when Christ died and was resurrected. Where was Peter at the time? In Jerusalem. Therefore Peter became the leader of the whole Christian religion and leader of the Christians in Jerusalem at the same time. So it can be argued that the Patriarch of Jerusalem is the rightful head of the Christian Church instead of the Patriarch of Rome.
It may be noted that Patriarch of Rome was considered first among equals, being the most prestigious of all five patriarchs, but the other four patriarchs acknowledged little or no papal authority over them. The five patriarchates were independent with no common authority over them in the opinion of the other four patriarchs.
And the supreme authority in all major theological disputes was considered to be ecumenical councils where "all" of the bishops of the world or their representatives gathered to vote on theological disputes.
Naturally different Christian denominations differ on which councils they recognize as binding. But the present day Roman Catholic and Eastern Orthodox churches agree on the first seven ecumenical councils from the First Council of Nicaea in 325 to the Second Council of Nicaea in 787.
And what bishops called those ecumenical councils? No priest or bishop called those ecumenical councils. They were called by emperors ruling the eastern part of the Roman empire. So if there was a theological issue dividing the church, your emperor would have the right, duty, and power to convene an ecumenical council to vote on the question.
And it may be noted that eastern emperors sometimes appointed and deposed patriarchs of Constantinople, and also patriarchs of Rome in earlier centuries.
So an alternate history where a Roman emperor having his capital in
Constantinople is the mighty secular ruler of all Christian lands (and perhaps of many non Christian lands), and where the Patriarch of Rome doesn't dare to challenge his secular power any more than the Patriarch of Constantinople does, and doesn't cause much dissension over theological issues, is easy to imagine.
A few historical events have to be changed:
1) The Plague of Justinian which killed many millions of people inside and outside of Justinian's Empire must be prevented. This would prevent the biggest disaster which weakened the Eastern Roman empire the most.
2) The Extreme Weather Event of 535-536, and the world wide famine and deaths it caused, must be prevented, even if it means preventing volcanic eruptions and/or diverting comets or asteroids.
With those two disasters eliminated and the eastern part of the Roman Empire much wealthier and more powerful in the rest of Justinian's reign, he should be able to reconquer the rest of the western part of the Roman Empire relatively easy.
3) To make the reconquest of the rest of the western part of the Roman Empire easier, Totila, King of the Ostrogoths in Italy from 541-552 must be eliminated, preventing his bloody and devastating resistance to the Roman reconquest, making Italy a much wealthier province.
4) The Avars must be prevented from settling in Pannonia (modern Hungary). This will prevent the Lombards under Alboin from emigrating from Pannonia and invading and conquering much of Italy, thus eliminated most of the freedom various popes had to disagree with the emperor. This will also prevent The Avars from making devastating raids into the Balkan provinces of the empire, and prevent their Slav subjects from invading and settling in most of the Balkans, taking them out of the empire for centuries.
5) Some sort of permanent peace with the Persians should be made. All of the eastern provinces in Asia, and some of them in Africa and Europe, were once part of the Persian Empire for 200 years during the Achaemenid Dynasty. The Romans made most of those regions provinces or client states by about 200 to 100 BC.
So by Late Antiquity/Early Middle Ages those regions would have been ruled by the Romans for about 400 to 800 years, and thus for two to four times as long as they were ruled by the Achaemnids.
So the Roman Empire could propose to make the Persian King of Kings a co emperor, perhaps in exchange for the Roman Emperor becoming a co King of Kings of the Persian Empire, and propose that the revenues of all Roman and Persian provinces which have ever been part of both the Roman and the Persian empires be divided according to how long each region was part of each of the empires.
Thus the Persian King of kings would be acknowledged as a co ruler of a large part of the eastern Roman Empire and would gain a vast amount of revenue from it, while the eastern Roman Emperor would loose a part of his revenue but gain the prospect of peace with the mighty Persian Empire.
And perhaps to make up for lost revenue the eastern Roman Emperor could use the monsoon winds that Graeco-Roman traders used to sail across the Indian Ocean to India to invade and conquer a part of India to tax.
Anyway, if some plan to make permanent peace with the Persians succeeds, it will prevent the bloody and destructive Roman-Persian wars, and the last and most terrible war of all from 602-628, which left both empires weak and vulnerable. Thus that would prevent:
6) The Arab conquests beginning about 630. If the Roman and Persian empires are both strong and powerful and on friendly terms the Arab conquests can probably never succeed and the unsuccessful invasions will probably cease soon.
Thus those are the changes likely to make a fictional emperor of the eastern section of the Roman Empire so powerful that no patriarch of Rome will dare to challenge him in political matters or make much trouble about theological ones.
