I kind of get the same, really. The "restored Byzantium" wouldn't be the old empire in any meaningful way. Not even in name: I'm pretty sure Catherine would never allow it to be a rival imperial crown. She'd make her grandson Constantine "king of Byzantium" or something like that. The key point is that her long-term desire was to unite this Byzantium with Russia. Then it would be an empire again.
Historically, her whole plan held about as much water as a sieve. But when you start looking at it as a long-term scheme to forge a Greater Russia, it suddenly makes a lot more sense.
The Tsar would probably adopt the title of Emperor and Basileus of the Romans, on top of his other titles. The Orthodox and Christian character would be strongly emphasized and the "Third Rome" idea would probably gain new emphasis as well, which would be used especially for political propaganda both internally and externally. I bet that they would even have children born in Imperial palaces designated as "porphyrogenetos."