What about the mob, surely they'll have some say in this?
Or is the populace of Rome too apathetic by the 480's?
I'm really wary about the XIXth trope "Romans became apathetic and so uncitizens". What happened was a political shift from elites that were more tied to the imperial power and then translated to the Barbarian legitimed imperium that eventually take their cliens with them.
And for what matter Italy, remember that the armies that Odoacer led were the same Roman armies than during the WRE, and Odoacer a Roman general. You won't have much opposition based only on the novelty of not having puppetised emperors (something generally seen as degrading for the purple).
Overall, you had more a focus for local and "everyday" management, even in Rome, based on strong ties with ERE but refusal of imperial interventionism : if the papacy (something reserved to roman nobility) is to be a clue about the relationship between Romans and Barbarians in Post-476 Italy, anyway.
The Roman factions would probably be divided, generally supporting candidates "appointed" by Constantinople as long the emperor doesn't try to impose his policies in the peninsula too strongly.
In this case, it would mean a relative desinterest : not meaning a general political desinterest, but rather attentism in face of their contradictions (support of Romanity, but wariness towards ERE).
I'd think that, safe a Zeno's move as IOTL (Acacian Schism) or obvious ERE tentative of puppetisation, "Constantinople's candidate" would be favoured but not outright supported.