Wait, so the republic was an actual democracy? I was under the impression that the Senate was basically a body of nobles.
It was no such thing. Of course, 'nobles' doesn't describe it very well, eitzher, but it was an assembly of the privileged and powerful.
The problem, with the OP is that we first need to define what we mean by 'Republic'. It would certainly be possible for the Roman state to revert to sdomething like the post-Sullan form of government at some point in the first century AD, with the privileged classes actually competing for meaningful power. I don't think that would last long because civil war between Roman generals always tended to be awfully hard on everyone not a general. But it would be doable. A pre-Sullan system of traditional checks and balances - probably not so much.
To the Romans themselves, of course, the question doesn't arise. They always lived in a res publica - Augustus and Valerian were both addressed as saviours of the Republic in their time. There was no transition to imperium, imperium was something you had, not something you were. You could have constitutional changes, of course, but the state did not stop being res publica because of trivial stuff like that.