Good question. I'd think that without a sudden quick decline of the ERE, as per rise of Islam, the Empire would remain a hodge-podge of competing Christian doctrines as would areas outside of Roman control, in the Persian lands. There is a lot to be said for the existential threat of Islam forcing Western and Eastern Christianity to become a lot less tolerant of dissent
Ah, but by the late Lombard period, you already saw the Pope declaring himself the ruler of Rome and its environs. Although the papacy might remain weaker with a strong ERE, i really don't see it not becoming the dominant religious force in Western Europe. There might well be challangers, and it may not be as strong, but Rome will still be dominant.
If nothing else, you may see a *Charlemagne figure emerge who decides to use the pope the same way the emperor uses the patriarch.