I found Byzantium pretty weak personally. Once you deal with their initial army spam Constantinople will be easy to capture and its over for them by that point. But I can imagine they become a problem if you don't bother them for a long time.
Even though I like the script, the better way to help Rum would be with units. I think the common horse archers should have normal formation instead of the tribal one, it would help a lot with micromanaging the army. I know they are "tribal" but they were still disciplined so it would make sense. If you think this would be OP, I would suggest taking them off of other factions' roster like the Abbasids. Another thing is I think an event should trigger if the Rums make it to Europe that allows them to have the non-gun Janissary units and the "Ottoman Infantry" (should be renamed really) unit earlier. This would help a looooot and make playing them more fun. I don't see any reason why the non-gun janissary units have to come in with the gun reforms. On my last run I got bored with the save before I ever had janissaries.