I added in a "Roman" culture under the Byzantine category, which I justify by assuming that:
1:The Latin-speaking Vatican priests stay in the Vatican, as all of the knowledge and books there are in Latin. Greek speaking priests would have been useless to replace them with.
2:That as a result the Roman clergy would remain largely Latin speaking. The Ecumenical patriarch may be unlikely to want to leave the Hagia Sophia and so his Greek wisemen would mostly stay in Constantinople (assuming you have moved your seat to Rome) - he doesn't anyway, as the patriarch of Rome is responsible for the West
3:That your children would be brought up bilingual anyway, as they get taught by Latin priests and Greek nobles.
4:That the Italian lesser nobility would remain. Yes, Greeks would rule the roost, but property would not be confiscated because people spoke Italian, hence there would be a language divide between Upper nobility and lower nobility.
5:That ultimately, the Emperor may decide that Latin will be the Lingua Franca between Italians and Greeks, because IT IS the true Roman language (despite what absurd Byz fanboys say), as it was the native language of actual Rome: the city itself. Due to it's almost mythical reputation and presence in the oldest of Roman records, the Emperor may demand more respect by speaking it whenever possible. He would encourage Greeks and Italians to both learn it (a language they can both accept as more Roman than their own and therefore get behind).
6:Latin therefore becomes the official language of the Roman nobility (although the lower classes still speak Greek, Italian and various other regional languages.
I mean why do scientists use Latin today? Why did Napoleon use Eagle banners? Reverence for the language and culture of Rome.