I see no real reason why it should be called Byzantine Empire in the game
No reason at all? I don't deny that calling them the Roman Empire or something along those lines would be correct, but I also recognize the term Byzantine Empire. And while some of their contemporaries did refer to them as Romans, others referred to them as Greeks. Especially from the time of Charlemagne the West would refer to them as the Empire of the Greeks. Perhaps that is what they should be called in game, since that is how they were referred to by some. I'm being a bit hyperbolic, since I don't actually want to see them referred to as that, but I hope it gets that point across, that how they were named by contemporaries doesn't translate to how we should call them in the game.
I think that if the game starts in 1066, calling them the Roman Empire would take away from the political controversy that is being the successor to the Roman Empire. While they are of course legally the Roman Empire as they have this clear line back to the Roman times, I think that spiritually, the "Roman Empire" has ceased to exist. What I mean by "Roman Empire" is the intercontinental empire spanning Europe, Africa, and Asia.
That's why I like how it is in CKII right now. The Roman Empire doesn't exist in the start, it has to be reformed. Not calling the Byzantines the Roman Empire makes reforming the empire that much more meaningful.
But that only address why they should not be called the Roman Empire. Why should they be called the Byzantine Empire? Well, if Roman is out of the question, why not call them their other name I mentioned, Empire of the Greeks? This name I feel doesn't capture that Roman Legacy that Byzantine does. Byzantine, we know it's Roman just by a different name, calling them Greeks is a rejection of Romanness and an embrace of Greekness that I don't think represents the situation. Byzantine is probably the most appropriate because, while anachronistic, is the modern English language term used to refer to them that represents their Roman heritage and their fall from that.