First things first
I would like very much to read an alternate history book where, in an attempt to preserve England for the self-titled Universal Church, Pope Clement VII steers an Oecumenical Council towards pronouncing that polygamy is permissible.
I would especially love a book where the doctrinal and temporal machinations needed for such a radical pronouncement would be explored in detail.
This being said, there are massive problems with the suggestion.
Why is polygamy not allowed in Christendom?
Seen from a great distance, Christianity is a particularly non-observant Jewish sect. The original founders of the Christian religion, notably Paul and the anonymous author of the Luke-Acts duology, were undoubtedly Hellenized Jews, who were not aware that they were founding a new religion. Is it uncontestable that Christianism began as a movement to reform Judaism and open it to the world.
Now, at the time when Christianism was emerging from the religious debris resulting from the fall of Second Temple Judaism, the Jewish religion had no problem whatsoever with polygamy; it was the Roman authorities who had a big problem with polygamy and insisted on men having at most one wife at any given time. And let's not forget that the original founders of Christianism were Hellenized Jews, and the Hellenistic world had inherited from the Greeks a strong presumption that civilized men did not have more than one wife at any given time. Sure, they could have more than one wife, but only serially, not simultaneously.
In short, when it all began the equation was simple: civilized men are married with at most one woman; being married with two or more women at the same time is for barbarians.
Christianism was trying to take over the people of the Empire; and the people of the Empire were civilized; and civilized men did not take more than one wife at the same time. Not only they didn't: they couldn't. The law was very strict. It was unconceivable to preach a religion which did not emphasize monogamy. It would have been perceived not only as utterly alien, but as fundamentaly barbarian; and one thing the inhabitants of the Empire agreed on was that they were not barbarians.
By the time of Henry VIII the Church had accumulated a millennium and a half of dedication to monogamy. Reversing its stance would have required a massive dose of radical thought, something for which Pope Clement was not known.
And then there are the practicalities.
First of all, in the times of Henry VIII, the Popes did not have unquestioned papal infallibility; this doctrine was promulgated at the First Vatican Council in the second half of the 19th century. Sure, there were some medieval theologians who argued that Popes were infallible under certain conditions, but papal infallibility was not a black-and-white doctrine of the Church. Any Pope who would even contemplate the idea of allowing polygamy would have been certain that the Church would splinter and lose both people and territories to the Protestants and the Orthodox. The only non-insane way to declare polygamy permissible in the eyes of God would have been to summon an Oecumenical Council, and work to get a consensus.
Second, we must dismiss out of hand the idea that the Church would even attempt to "doctrinally permitted for people of great wealth [...] to be married to more than one woman at the same time". It is a basic doctrine that all men and women are equal in the face of God; stating that rich men are more equal than poor men would have been anathema in the 16th century as it would be today. This cannot possibly work. Either all men can marry more than one wife, or none can.
Third, there is the problem that all men and women are equal in face of God. Having one rule for men and another for women would have been quite iffy; borderline possible, but practically guranteed to raise a lot of opposition at the Council.
Fourth, such a decree would have placed canon law in stark opposition with the secular law of all Christian countries. For that millennium and half during which the Church had placed itself firmly on the side of monogamy had had the effect that the secular laws of all countries banned polygamy, had strict punishments for polygamous perverts, and did not even consider the possibility of people having more than one spouse. After all, king Henry was not concerned much with the sinfulness of having more than one sexual partner: what he wanted was a legal heir.
Fifth, the Church would have had to innovate and emplace a new form of marriage ceremony. As inherited from the Romans, Christian marriage ceremonies emphasize the equality of the groom and the bride and require freely given consent. The new form would have had to insert one or more third parties, namely the pre-existing wives, and devise a form for all the parties involved to express their approval and to take the respective vows. (The pre-existing wives need must be part of the ceremony because they are to accept the new wife as a sister and take vows to live in peace; a vow is not valid unless freely taken.) Having the Catholic Church devise a radically new form of marriage ceremony is not a task for the faint-hearted.
The tactical situation
Pontifex Maximus is a great and ancient title, and it confers great power. Except when it doesn't.
The tactical situation was that at the specific time in question Pope Clement was not a free man. In 1527, the soldiers of the combined empires of Charles V had sacked Rome; the Pope himself had escaped disguised as a humble peddler. When the petition of king Henry VIII for the annulment of his marriage with Catherine of Aragon reached the Pope, Clement was for all practical purpose a prisoner of the emperor. And Emperor Charles V was the nephew of Catherine.
Emperor Charles V reigning resplendent over his vanquished foes; pope Clement is second from the left, standing next to sultan Suleiman I of the Ottoman Empire. 16th century allegory. Reproduction available on Wikimedia, in the public domain.
Both the Most Catholic Emperor and the Catholic Church were also much preoccupied with the issues resulting from the Protestant schism which was making inroads in the Germanies. By allowing polygamous marriages they would have given the Protestants direct proof that the Church of Rome was truly the Whore of Babylon, possibly with the immediate consequence of pushing more German princes into the welcoming arms of the Schmalkaldic League.
To sum it up
The Pope was not able by himself to make polygamy permissible. He would have needed the consent of an Oecumenical Council.
The Pope was under the thumb of Emperor Charles V, the nephew of the spurned wife Catherine of Aragon.
Both the Emperor and the Pope had much more pressing issues on their mind than the matrimonial situation of king Henry.
Even if they had considered it, it was not a simple thing to do. Lots and lots of changes would have to be effected both in canon law and in the rituals, entailing a massive risk of splintering the Church.
And even if the Church would have been willing, English law was not ready for it.