Nope I did a paper on Napoleonic Egypt, the foundations of his defeat began before he began.
It was pretty much doomed from day one, and a lot of people suspect that the Directoire picked Napoleon to do it exactly because it was doomed (his political meddling was becoming inconvenient). Plus getting hit by one of the last major plague outbreaks in Alexandria was another layer of rotten luck...
As for the OP's question he would probably have picked something more modern, like Sultan of Egypt IMO, and it was before the empire, the egyptian campaign of 1797 would likely have led, at best, to an Egyptian Republic (hopefully not a Nilotic republic like all the italian subrepublics before 1802 were called by their position on the Po or the Alps, yuck).
For an entertaining POD; Napoleon is too succesful in Egypt and the directoire basically dumps him there indefinitely; Bonaparte-ruled sultanate of Egypt anyone? (yes, I know, probably ASB)