I had in mind of keeping the Legion and Bath but making the alt leader, either Justinian, Constantine, Theodora etc. give a completely new UA as well as part of their own UA. Call it "New Rome" and let that ability add another UU (Dromon) or UB (Basilica, Hippodrome).
Constantine the Great wouldn't be that bad choice for a Roman leader, but the only thing that makes him connected to Byzantium is, in my opinion, the fact that he founded Constantinople (or, well, rebuilt and renamed the already existing town of Byzantion). He was a sole leader of the Roman Empire, and still much more of an Imperial Roman leader than a Byzantine/Eastern Roman one. And Imperial Rome is a period that is already perfectly captured by Trajan, one of the best emperors in Roman history.
Now, Justinian and Theodora have been around for three games already. Justinian may be a very notable Emperor and Theodora a capable co-ruler, but heck, let's not pretend that their era is the only glorious and worthy period in some 1000 years of history that Eastern Rome/Byzantine Empire has to offer. They weren't flawless, either. While I do not deny their reign bringing many things (Hagia Sophia, Corpus iuris civilis, conquests, defeat of Níká riot), but let's not forget that Justinian had full treasury in the beginning, a luxury many of later Byzantine Empire didn't have the luck to begin with, was disliked by his people, his campaigns were mostly succesful thanks to having Belisarius and Narses around and that his famous conquests had a little unfortunate side effect - they managed to make the Empire as rich as Eastern Rome ruined financially. I think emperors who had to fight against great problems during their time and being successful, even leaving the empire stronger and with full treasury would be nice. For this, Basil II Bulgaroktonos (peak of after-Justinian Byzantium) and the first three Komnenoi emperors (Alexios I, Ioannes II and Manuel I) are ideal choices. I wouldn't even mind Michael VIII Palaiologos, who managed to restore Byzantine after it was wiped out of the world map by the Fourth Crusade.
And if we really need an "early" Byzantine/Eastern Roman emperor, we could go with Heraclius this time. Yes, I am aware that during his reign, the Arabian expansion robbed the Byzantines of the riches of Levant and Egypt, but it wasn't his fault that the Muslims came when the Empire was exhausted by twenty six years of war against Sassanid Persia. Thanks to him, the war ended in the first place, with the Byzantine borders surviving the war intact despite the fact that the Sassanids occupied most of Byzantine lands in the East and even reached Constantinople, and unlike the Sassanid Empire, the Byzantine Empire didn't collapse entirely to Arabian expansions, but actually managed to live (and sometimes even thrive) for more some 800 following years.
There are great possibilities Byzantium has to offer, and I feel like they are better to be explored through a standalone Civ rather than through an alternate leader of Rome.
And, as I said, I want Byzantine chants back. Give me them space dromons, Varangoi and kataphraktoi!