There are mentions in the souirces at various points of Byzantine empresses considering marriage with western rulers, but mostly these sound unlikely to me. However, if you wanted to run with it, the alleged marriage proposal by empress Irene to Charlemange could work as a fictional POD. Contemporary Byzantine writers considered him a barbarian upstart presuming to claim a title he had no right to, so the popular reaction in Byzantium would likely have been strongly negative. Still, Irene, in need of some military muscle to beef up her shaky hold on power in the middle of the iconoclastic conflict might have allowed armed Franks to run riot through the empire. For a while. Still, top get a lasting union would need a more conciliatory approach on the part of both sides. I think to get that you would have to remove the papacy. or at least install very pro-imperial, un-Gregorian popes (still possible at this stage. If Pippin the Short or Charles had lost their tempers with the popes they could simply have swallowed up the Patrimonium and sent the bishop packing - but they didn't).
After that date it's highly unlikely. The Frankish inheritance rules mandate a division of the kingdom, so simply marrying a Frankish princess would not secure a Byzantine pretender the throne. Besides, there is the Salic Law - not held to stricvtly, but certainly remembered in such an instance by the aristocracy.
There is an interesting point at which Otto III, the young son of Emperor Otto and the Porphyrogenneta Theophanu, could have become heir to both empires, had he lived. He is one of my favorite PODs as a kind of 'enlightened ruler of Christendom', but in reality I don't think the gaps could have been bridged at that point. Western church culture had gone too far adrift, and the rivalries were too sharp already. A half-century later you get the Normans, and wonce they're through with Byzantium there's zero chance of reconciliation.