Court and government are two different things
The Royal Court and His Majesty's Government are two very different things, which serve different functions and are made up of very different people; so that, well past the time when this was what was expected of a monarch, the king can choose an itinerant lifestyle if he so pleases, or if he so must.
The times of the stereotypical itinerant monarchs belong to a historical period when there was not much, if any, in the way of central government. Charlemagne could obviously spend all his reign travelling from place to place, because he ruled during the most medieval times of the Middle Ages, when even the notion of a central government was alien to the realms of Western Europe; he simply did not have any kind of government, and the idea that he (or the people immediately around him) would be expected to actually govern the empire did not even cross his mind. The empire was made up of great fiefs, and it was the business of the holders of those fiefs to govern them; the emperor had other business, such as to make war, to hunt, to sire illegitimate children, to make merry, to receive foreign delegations and, time allowing, to attempt to set policy.
But even before the Middle Ages, in the days of the Roman Empire, some emperors choose the life of a perpetual tourist; Hadrian comes to mind: he could be found anywhere in the empire but almost never in Rome. Other emperors had to live on the frontiers: for example, Trajan and Marcus Aurelius. Yet others, such as that famous pervert Tiberius, had a fixed abode outside Rome, and expected to be left in peace unless some important decision had to be made.
Unlike the medieval Carolingian Empire, the Roman Empire actually had a government; but the physical presence of the emperor was not necessary for the operation of the government. They had senators, and quaestors, and praetors, and praepositi, and procuratores and so on to take care of things.
In post-medieval Europe we also find examples of such emperors and kings. Consider the magnificent Charles V: he was both Emperor of Spain and Holy Roman Emperor. Both the Spanish and the Holy Roman empires had governments of sorts, in Madrid and in Vienna, but those governments could, and actually had to work without the double emperor holding their hands. Charles V travelled extensively throughout his European realms, as the needs of war and civil and religious strife required. (He did not like it much; at the age of 56, after 40 years of service, he abdicated both crowns and went to o a monastery.)
Or consider the Sun King, Louis XIV of France. He and his court were fixed at Versailles throughout most of his reign; yet the government of France remained in Paris, and for three decades both parties found this arrangement to be perfectly acceptable.
And even in our up-to-date modern days, I understand that American Presidents tend to be grabbed by an irresistible wanderlust when elections approach; for example, in October 2018 President Donald Trump of the U.S.A. have no less than 18 speeches at events all over his huge realm; yet the government of the U.S.A. stayed put in Washington, D.C., and governed the country, and felt no inconvenience.