Greek Empire would reflect it and be actual name used by western world, Byzantium/Byzantine Empire is wrong in every way especially during 5th century.I think "Byzantium" could be handled through them starting the game as "Eastern Roman Empire" and if they fail to reconquer enough provinces to reform the empire their name will be changed to "Byzantium" to reflect the empire being considered the empire of the greeks rather then the true roman state.
I don't understand why anyone would ever call the Roman Empire the Byzantine Empire, fact they changed their focus from Latin to Greek is irrelevant, it was still based on text written in Latin. These Byzantinophiles are stubborn people who have been misguided, they have no logic, nothing to back to their silly claim that makes no sense, yet they refuse to see the truth.