Other than "Ave", "Vivat" ([long] live) comes to mind.
I have also found that in a toasting-cheering context, the proclamations "dii propitii!": (May the Gods be propitious!) and "Feliciter!" (be lucky, happy) were used.
The Wiki article makes quite a good job of demonstrating that "res publica" was used to refer to many different concepts:
Tacitus did use it to signify the political system of the Republic, as opposed to imperial government.
However in many contexts it simply meant "public affairs".
It could also mean the State, even when under emperors.
So when uttering patriotic cheers, referencing the Roman Republic as a polity or nation, I think they would simply say Roma or Popolus Romanus.
When saluting the political system (for example to protest monarchy or tyranny) they might say "res publica", but I think it is more likely that they would prefer "libertas", "mos maiorum" (ancient custom), or some way of referencing the ancient constitution (the res publica wikipage claims without citations that they would mention the Twelve Tables in this role)