The newly converted Arab Christians could prove a valuable asset to the Byzantines against the Sasanids. After the fall of Persia and Persia becoming Christian, its fair game what their relationship would be to the Byzantines.
This of course is the obvious answer. The OP is moving in different time periods - Islam rose long after, and probably because of, the spread of Christianity.
Having studied the Visigoths and Vandals in some detail, it's not clear that their actual behaviour was in any way related to whatever religion they were following at the time. Both the Arian Vandals and the Catholic Visigoths conducted themselves in ways that would today be seen as 'not commensurate with Christian values'. But so did the Spanish and also the Pope so - hey, that's history!
Again depends on Persia. In real life the turks conquered Persia and adopted its culture before moving on to attack Byzantium. Even if they adopted Byzantine orthodox they would be influenced by Persia. Possibly as allies of Byzantium or rivals.