I think a strict geopolitism would be a superficial way to see the relationship between papacy and Latin churches.
Assuming a Byzantine takeover of Rome, which is clearly not that easy, at the very best you'd have a reverse Greek-Catholic church, a Latin-Orthodox church if you will. Probably more autonomous religiously as a mega-patriarchate.
I think that people greatly exagerate the opposition between Rome and Constantinople, as if both were somehow limited enough to act as Crusader Kings AI : in the case of a Byzantine reconquest of Rome, it's likely that the emperors would put someone there as a religious head for the West, being far less concerned to curb down Latin church, than making it "compatible" with their own cesaropapism (and would certainly not try to abide to radical eastern clergy demands, as Rome would be probably needed to balance their influence as it happened historically).
Roughly speaking, a return to pre-VIIth situation, except with a much more defined Latin church, with a Byzzie-friendly pope.
The real, and growing, difference between Latin and Greek churches never really went to the point they weren't porous enough to reach agreement on the institutional level (popularily, it's another story, but it was much more obvious in Byzantine population than for Latins).
As for reaction from western kingdoms : pontifical legitimacy was far too tied with its relation with Rome to really survive a permanent exile so far from it. Avignon Papacy was essentially a thing because Avignon was as much a pontifical territory than the Italian cities (as Perugia) popes often resided semi-permanently.
Either Papacy-in-Exile vanishes after a while, with whoever is set in Rome becomes the religious head in the west; either this papacy in exile becomes a peripherical schismatic movement without much relevance.