If you mean "the Carolingian Empire", the name itself, it's not a name anyone would have used. Charlemagne and his successors called it the Kingdom of the Franks, and later on, the Roman Empire.
If you mean the state itself, technically both the HRE and France are descendants of the Carolingian Empire. "Carolingian Empire" is just a modern name. If someone wanted to create his own Empire in Europe before the 1800's, they'd probably want to either take the HRW crown from the Habsburgs, or become a crazy Napoleon-type with no legitimacy.
Imperial tradition kinda died out during the Wars of Religion, and was picked up again in the late XVIIIth by the first Romantics.
But I'd like that the game would allow you a certain degree of leeway when creating yourself Emperor, for example. "Emperor of Saxony"... meh.
Leaving the names and requirements to be able to form it aside, I honestly think that It wouldn't be so difficult to implement in the game if they do what they have done with the kingdom of god or the caliphate. And as you said well, it would bring diversity to the game.