Why would God's Vice-Gerent on Earth abandon the religion that legitimises his authority in the eyes of both Church and (increasingly) the rest of the population? Constantine got away with it largely because there was a significant groundswell of Christianity that was popular enough that he could gain pre-eminence over the other tetrarchs simply by tolerating them. You need Zoroastrianism to become massively popular in the Empire for this to be even feasible. Several problems:
1) Zoroastrians don't really go into proselytizing. It was never their thing particularly. This is due to the fact that the religion was almost synonymous with the Persian state - a kind of Cuius regio, eius religio scenario avant la lettre. Two solutions: have the Sassanids take Constantinople, which I feel isn't really in the spirit of the thing; or have the ERE expand so much into Sassanid territory that there is a major demographic shift in favour of Zoroastrianism. Since the Persians clung to their faith well into the Islamic era, it shouldn't be too hard to prevent them from disdaining conversion to Christianity, but even then the Zoroastrians aren't going to try to convert many others without a POD in the BCs.
2) Christianity was particularly proactive in this era. The heated conflicts between Chalcedonians, Arians, Monophysites, Iconoclasts, etc. etc. show that a) the wanky disputes of bishops held sway over the State enough to make the conversion of the Emperor to a completely different religion kind of a big deal - a blinding matter, certainly. And b) the man in the street was devout enough to undergo persecution for heresy and/or practice state-sponsored destruction of icons. These guys aren't going to roll over and take a literal baptism of fire.
3) As the only way this is going to happen is complete Sassanid conquest of the ERE and a lot more besides, then Islam is going to have a field day against such a far-flung, over-committed, overstretched Empire that is trying to force half its population to join the state religion pronto. This is, of course, providing that Islam arises in the first place.
So I'm sorry to infringe on your vibe, but I don't see how this is all that possible. I'm more than willing to be corrected, though. I'm half-Iranian myself, and I'd have liked to have our major contribution to the world survive in some form less pitiful than today OTL.