This is completely wrong, the greeks of the Period called themselves Rhomioi/Rhomans,so if such a state existed Rhomania(land of the Rhomans)would be the default option by the natives
The term Byzantine surfaced a century after the Empire had fallen
It could be called Hellas
Though at this point the Term Hellene and Rhoman are still synonymous
You imagine, perhaps, that they would get much of a say. I very much doubt that. If the premise is that the Greek Plan goes through, what matters is what the Russians would want to call this state (and what other powers have to say about that). Note that the Russians -- and specifically Catherine -- explicitly used the term 'Byzantine' in this context. She specifically had a medallion made when Constantine was born, engraved with the phrase "Back to Byzantium".
Considering that the other power most likely to be involved -- and thus most likely to get a real say in the naming -- was Austria, this Neo-Byzantine state getting to call itself "Roman" (or some variation thereof) is not likely. The Habsburgs, after all, are still Holy
Roman Emperors at this point. Just as they'd be wary of letting someone else have the title "Emperor", they wouldn't be happy to revive a competing claim to the Roman heritage. They'd insist that the official identity of the new state would be explicitly Byzantine/Greek.
In addition, "Hellene" and "Rhoman" were only still considered to be synonymous by Greeks of the time. Nobody else had taken that notion seriously for centuries. You may be correct to the extent that the native population would default to calling their country "Rhomania", but my point is: that wouldn't be (allowed to be) the official name, and it wouldn't be used internationally.