There may be some advantage to this. OTL, both the Sassanids and the Arabs were able to exploit tension with Constantinople. I want to check something when I can get to a book, but I think Copts only threw in with the Sassanids after they saw that the Sassanids would be 'tolerant'. Shahrbaraz seems to have done a good job with the whole 'hearts and minds' aspect.
If you don't have the Sassanid occupation first, then the Arabs may need to take some of the same actions as the Sassanids did to weaken Chalcedonianism in Egypt. You could also follow OTL with Egypt going Byzantine->Sassanid->Byzantine->Arab, but without the Islam.
This strikes me as being a better plan, especially is the Byzantines are distracted by other conflicts in the Balkans, Anatolia, etc. Allying with Constantinople and then the native Copts seems to me to be more likely to antagonize both groups. Then again, a good, charismatic Exarch might be able to pull it off.
Do you intend for the Arabs to retain their religion at least initially?
The bolded part actually seems like a good plan, come to think of it. Maybe just having the Meccans win the early Meccan-Islamic wars might work as a POD. All the chaos in the Arabian peninsula results in Arab tribes invading-but rather than as one united religious group, it occurs gradually, over 620-700, with various Arab tribes coming into the Middle East, allying alternately with the Byzantines and Sassanids, and carving out their own kingdoms, much like the Germanic tribes and western Rome.
And specifically for Egypt, here's what I have in mind:
-Sometime in the 640's-650's, Syria has fallen under the control of various Arab groups, who have handed a couple defeats to the Byzantines. Egypt is still Byzantine, but the land connection to the empire is tenious at best.
-A group of still-pagan Arabs take over Egypt. Their chief, wanting to avoid Byzantine attacks, agrees to acknowledge Byzantine authority in exchange for the Byzantines appointing him Exarch of Egypt (I understand a lot of Germanic tribes made similar deals with the western Roman Empire at first). As part of this deal, he and his tribe agree to convert to Orthodoxy. However, this is mostly nominal, and in practice the Arabs see the Orthodox church and the Greco-Egyptian population as potential fifth columnists in the event that the Byzantines get any ideas about reconquest. So, the Arabs make a general policy of treating the Coptic church much better than the Byzantines had-cultivating the Coptic patriach and appointing secret Copts to administrative positions (and looking the other way on their secret Copticism)
-Though the Arabs try to keep up appearances for the Byzantine Emperor's consumption, relations gradually grow worse and worse, and the scenario proceeds like I originally described.