I don't think that Carthage would change it's name. Just as the Romans never diverted from SPQR, although they no longer were a city state. The reason is that the Karthadastim empire is totally focused on Carthage, since it is a city-state, similar to Rome (though very different of course). Empire without Carthage - no empire, just Punic colonies. Or do you change SPQR to Imperium Romanum?