It doesn't matter how burgeoning the economy of medieval Europe was, or how supportive the Italians were, shipping supplies and reinforcements thousands of miles on mediaeval boats was going to be a logistical nightmare no matter what.
Medieval Europeans beat everyone, most spectacularly when an eighty year old octogenarian sacked Byzantium's capital and carted away its treasure and a quarter and half a quarter of its empire. I wouldn't presume the Islamic States would have trouble beating Byzantium.
But Frankly, a Byzantine state that's capable of conquering Egypt (or conquering any territory for extended periods) is so different than OTL's Byzantium that it's really hard to say wha tit can do.