I think it'd be a very interesting scenario, the Papal conflict aside.
It introduces other potential conflicts. Whilst the Carolongian WRE and Byzantine ERE are both "allies", the Carolingians still have the Italian problem of the Alps - which the Byzantines may be loath to help them with, without being granted rule over the area instead. This may well strengthen Carolingian rule as their resources aren't as stretched, but also leaves a robust Alpine divide between the two halves of Rome.
Incidentally, this leaves two very interesting situations.
1) Potential war between the two halves of Rome over N.Italy after a revolt is put down by the Byzantines.
2) Potential guide of succession in the Byzantines. Charlemagne could legitimately appoint a single successor, who can have Byzantine backing, helping strengthen their position (again, the price of this might be Italy).
But with the Italian Question resolved, the WRE in the Carolingian Empire is in a position that it can either emulate the Roman system in time, and stay united (keeping Francia as one massive trans-Rhine polity), or continue to fall apart and become a series of Byzantine satellites, if not outright client states.