It's really hard to think of such exile being maintained for more than one or two generations : Papacy 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.
I think that people greatly exagerate the opposition between Rome and Constantinople, as if both were somehow autistic 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.
Roughly speaking, a return to pre-VIIth situation, except with a much more defined Latin church.
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).
