I read some months ago an entire book about Byzantine empire, but I either study really in historical field, nor I am especially aware of what Byzantine empire really was (in comparison with some of you). However, I think that the primary conflict, in all Middle age to modern day, was a religious one. And I see some "correlation" between the degree of conflict between religious and political powers and the tendency to be a laic country today. I think the Byzantine empire is one in which political power initially won against religious one, until it was crushed and, in the last year of this empire, religious like the Patriarch were of great influence, leaving Muscovy to be the successor of Byzantine empire.
The fact is that I would like the game to show us the difference between the Roman catholics and their threat of excommunication, which was always a Damocles sword fot kings, and the more centralized (but not absolutely, since Heraclius) structure of the Eastern roman empire, in which the Patriarch was clearly under the emperor control when the emperor was strong. In the muslim world, things were even more "mixed", because political leaders took entirely the religious influence. Maybe I'm wrong, thought.
Since this game is about the "Crusader Kings", I think what I'm saying is important. Dynamic was different. The emperor was not dethroned because he didn't listened the pope or the patriarch, but because there were clearly personal conflicts, which were not mediatized by the church. Maybe this is what lead XIX century historians to conclude that byzantine empire was such a conflictual world, with greed and anger.
Finally... What could I say about the game, given all of that? I think piety should be less important for eastern orthodox monarchs than it is for western linked to the pope ones. But this system should be dynamic and depend of the influence of religious centers, since piety is prestige for the religious.