Could the ERE ever inherit the HRE instead?
HRE is a bit different on this regard, and links did existed between two entities (critically with Ottonians policies being actually influenced by Byzantine unions.
You'd still have to deal with confrontational politics (in Italy mostly, but not only)
Theophano, the mother of Otto III, was the token of a prize treaty; as HRE and ERE were in frequent conflicts.
Unions there are doable, but not inheritance. Ironically, byzantine influenced policies are partially responsible : it made a new conception of imperial power appearing, strengthening some institutions (as Papacy) at their own benefit, increasing the rupture between East and West.
Furthermore, this stregnthening of imperial authority displeased the german dukes that never really lost an occasion to rebel (critically during regencies, and if the regency of Teophano can teach something, is that Germans didn't liked "Greeks" at all).
I would tend to think that Germans would simply refuse it, and prefer elect someone else.
Italians opinion could be a different thing, that said. If made soon enough, before Schism for instance, you may have an important enough pro-byzantine faction supporting a byzantine claim (without Byzzies actually supporting this claim, but using this support to expand their authority).
That Italians, even supported by Byzantium, would be able to hold Germans would be still very hard, and would have slim chances. Still, it's not unthinkable that at least this stand happen.
On what matter Byzantines.
First it would require unions much closer of the emperor's family. It's not impossible, as it did existed IOTL, but only after Franks (in the larger sense) intervened more importantly in traditional Byzantine sphere of influence after the Crusades.
You'd need Byzantines realizing that Franks could be valuable partners, or at least, that they're a threat important enough to be tamed. Unfortunatly, before the Crusades, that's not really obvious, even without counting the traditional reject of Franks and Latins that existed.
The best window of opportunity for such would be during Manuel I reign, or an analog reign; but at this point, Latins and Greeks are estrangered enough that it wouldn't be really viable.
Summarizing it, I could see a byzantine emperor, or more likely a co-emperor tasked with it, using a claim to enforce imperial authority on Italy rather than trying to enforce it (a bit like Plantagenet stretegy in early phase of HYW).
That would ask for important changes before it can happen, and would be unlikely eventually, but it could work.