Some ideas stemming from the discussion:
1. The primus inter pares among the five patriarchs winds up being Alexandria, Constantinople, or, much less likely, Jerusalem. Alexandria is the most likely contestant but it means Alexandria winning the fifth century doctrinal wars and that has significant butterflies. Constantinople would produce some butterflies because the Eastern Empire would have to be more powerful and more involved in Church affairs, when IOTL it was probably as powerful and involved in Church affairs as it reasonably could have gotten.
Also, its hard to keep Jerusalem in Christian and Roman control, against the Persians as well as the Arab Caliphate. Alexandria is more doable, especially since if it wins the 5th century doctrinal controversies Egypt would be a center of the empire and the Egyptians more supportive of the empire (who lost the province to the Persians as well as the Arabs) but again you get massive butterflies.
Then you have to have the dominant patriarch in Alexandria or Constantinople create a second patriarch in the West and to somehow avoid the church in the West from just splitting off. IOTL Constantinople and the Byzantine Empire, which was on the upswing in the 10th and early 11th centuries, getting more powerful was a major reason why the Western church separated from what became the Orthodox church.
2. Constantine or one of his successors decides, or is convinced by his advisors, that the organization of the Church should mirror the administrative organization of the late Roman Empire more than it did IOTL, and that each Praetorian Prefecture should get its own senior Patriarch. This is a more promising approach than the first option.
The late Empire was divided into four prefigures, from west to east Gaul, Italy, Illyricum, and the Orient. Of these, Illyricum was the smallest and least important and seems to have been created last. It happened to be the main recruiting ground of the Roman Army. The Orient encompassed most of the Eastern Empire and contained four of the five senior patriarchies, and had by far the highest population of Christians in absolute and percentage terms. Originally the Eastern Empire was co-extensive just with the Orient, but when Theodosius re-divided the empire among his sons, the elder son, Arcadius, got Illyricum as well and that may be when a separate prefecture was created for the area (most of the Balkans). The efforts of the Western Empire to get it back complicated the imperial response to the Goths.
In addition, the two official capitals of Rome and Constantinople (the Western Emperors were usually resident at Milan or later Ravenna) retained their own Prefectures.
In this timeline, the Emperor intervenes and arranges for each of the Orient, Italy, and Gaul to get its own Patriarch. Illyricum is not created yet and had a minor patriarch anyway at Salonika. Rome and Constantinople still get their own patriarchs. ITTL, the Patriarch of Jerusalem is not created and Antioch is either relegated to minor status or given jurisdiction of Syria, Mespotamia, Armenia, and Christians in the Persian Empire exclusively as a sixth Patriarch and is removed from ecclesiastical politics within the Roman Empire.
So in addition to Rome and Constantinople, plus maybe Antioch as described above, Alexandria makes the cut as the Patriarch of the Orient (you could also make the schism with the Copts worse and have Antioch be the Patriarch of the Orient). Patriarchates are created for Italy and Gaul, most likely at Milan and Lyons. Now you have there Patriarchs in the West, at Rome, Milan, and Lyons, and three in the East, at Constantinople, Alexandria, and Antioch, perfectly balanced. This is due to an imperial intervention, for administrative convenience, so the distributions of Christians at the time doesn't matter.
If the Caliphate arises on cue and takes over the same territories, well now there are three Western Patriarchs and Constantinople is the only Patriarchate in eastern territory and is much more important. And the Ottos and their successors may opt to favor Milan as the senior Western Patriarch instead of Rome. The Kings of France will try to boost the status of Lyons.