/u/Arcvalons had a similar question about six months ago, but unfortunately no answers actually addressed the Byzantines.
I know that the Byzantines saw themselves as a continuation of Rome and as Romans themselves, but I don't know much more about how they looked back on their history, particularly the Republic. Was it idealized? Did people want to go back to that form of government? Were they even aware of the Republic?
Someone with better knowledge of Byzantium than me will hopefully be able to provide a better answer than this. But I can at least say that yes, they certainly were aware of the Republic. Many of the sources we use for studying the Republic today are in Greek, and survive thanks to Byzantine copyists. There was a Byzantine scholar called John Lydus who wrote a (fairly inaccurate) work about the magistrates of the Roman Republic. Then there's the massive Byzantine encyclopedia, the Suda, which includes entries relating to the Republic.
The "Byzantines" saw themselves as Romans not Byzantines. The term "Byzantine" was invented centuries later for the convenience of historians. "Ῥωμανός" (Romanos) is still means Greek [cf https://en.wikipedia.org/wiki/Romanus].