The religious differences were indeed a thorn in the side of Byzantium, but far less it was before : the imperial attempts at covering the difference trough monothelism (basically, a doctrine that focused on the energy rather than the nature, putting most discussion under the rag).
With a weaker ERE (maybe defeated by Persians, or forced to take in account Latin conceptions on this regard), you may have monothelism have an harder time appearing, if at all, probably.
But even without this, the differences shouldn't be that exaggerated : while Coptic elites have been relativly hostile (mostly because an imperial takeover of their traditional sphere of influence and power meant a loss of political and cultural relevance), the whole population didn't exactly jumped in the arms of Sassanids (if admittedly, Egyptians didn't exerced that much resistance either).
And that's going to be the main problem : forget about Arabs, Persians are there and ready to conquer Egypt as they did IOTL.
Without Byzantium support, Egypt have even less chances and would be conquered.
Depending on how the Roman-Persian wars go, you'd have either the region remaining under their rule, or Byzzies reconquering it.