So historians are quick to point out that the Byzantine Empire is the continued line of the Roman Empire under a different name, and all the complexities therein.
But let's assume that plagues, famine, and other such calamities had not befallen the Eastern Romans in their march through Italy and their attempt to reclaim Roman territories.
The public view now is currently, it seems, that the Byzantines were Rome Lite, or Diet Rome. If the Papacy had not been able to crown Charlamagne and help establish the Holy Roman Empire, if the Byzantines had seized most of Italy as was their goal and held onto it, would that have bolstered their claim enough to echo back to us?
I suppose what I'm really asking here is, did the Papal attempts to stifle the Byzantine claims on Rome have a significant impact on our modern perception of the Empire, or would we view it much the same way today if the Papacy had been brought to heel?
Claim to what, exactly?
I think you're asking an interesting question, but in a slightly confused way. Furthermore, this is really a bit more down the /r/HistoricalWhatIf line - what would have happened if Byzantine forces had held Italy?
Part of the course of actual history is the development of European identity out of the territories of the former Western Roman Empire, on the basis of latin language legacy, common Catholic religion, etc., and in that milieu the Byzantines came to occupy the place of the 'other', the strange, exotic east (ironically, this was the role played by the Persians to the Greeks, now being played by the Greeks to the West). In my view this is part of the construction of Byzantine Identity in Western eyes.
But if Byzantium had held Italy, the whole course of European history would be altered in so many ways that I would have to write a spec-fic novel to explore them.
tl, dr: it's hard to know what counter-factual history would turn out like.