It's worth mentioning that most of the cultural/theological differences that came to a head in the Great Schism were already present before the fall of the Western Empire. The theological underpinnings of the filioque had already been articulated by Augustine. The liturgy was in Latin, not Greek, and iirc was celebrated with azyme rather than leavened bread. Popes like Damasus were already claiming a form of papal supremacy, and again you have statements from Augustine like "Roma locuta est, causa finita est."
A Western Roman Empire will have an interest in holding onto as much power as it can and to maintain its legitimacy in the face of Constantinople's growing supremacy. It will have an interest in protecting the Church of Rome's prerogatives as it sees them as a means of maintaining control over its own people and as leverage over the ERE should it need it.
However, the manner in which the Western Empire survives will have an enormous effect on the theological controversies which follow in the proceeding centuries (I think it's fair to say the Christological controversies would take a very different shape, and that Pelagianism may not be as successful if it arises at all). It would be difficult for me to forecast how the fault lines would develop/be exacerbated without more information. Who knows? Maybe the Western Emperor becomes an Arian to appease his Gothic allies and the Schism is both earlier and more dramatic.
There are a couple things that would help prevent a schism though. One is that a surviving WRE is much more likely to have continued access to Eastern writers and vice versa. A maintained theological dialogue might prevent some of the linguistic misunderstandings that only raised tensions. The other is that an emperor in the West, and in Rome herself, might not want the pope to be too powerful, otherwise he would run the risk of not being able to control the machinery of state.
These are just some of the issues as I see them. I'd be interested to see where others take it.