Maybe Latins and Caesarians or Imperials. The Empire was lead by an Emperor who was always called Caesar. SPQR is also an option but not really a civilisation name. Though neither are the 3 Kingdoms. And technically the Inca are also just the rulers and not the people.
But I personally wouldn’t mind the much simpler 2 word civ names “Roman Republic” and “Roman Empire”.
I made a concept for the Romans that has both the Republic and the Empire as separate civs but also as combined civ that even includes the Kingdom.
Chronicles: Roman civilisation concepts
I should update the concept a little and add some more interesting mechanics like a Senate.