In my opinion, If one of the "Empires" conquers certain provinces and unites all Greek factions it should be renamed to Byzantine Empire and should get access to all units like Varangians, Cataphracts, etc. Because it will be somewhat weird if Trebizond Conquers all the old Empire lands and won't move capital to Constantinople or the emperor won't call himself Emperor of Byzantine Empire instead of Emperor of forgotten province Trebizond. 
Well i didn't read the deleted posts and idk what was going on here, i will just write my opinion and some facts. In later period, with the influence of Mongols, Georgian Nobles adopted bows as secondary weapon. They still preferred mounted shock combat, but against certain armies like the mobile Turks they started to use the bow. They didn't have lances attached on back, instead they were holding them for a while before breaking them after the charge. They didn't have bows attached on their backs either. Both would be quite impractical. It was attached to the horse.
I like the units with both lances and bows, but after switching the lance to bow, they should drop it, i guess it should just disappear. And the bow should go back in special sheath on the horse. Is this possible?
Also, can such units also have swords? I guess no.
Here is one good example, just change the ax with long lance.