With a POD no earlier than the Pyrrhic War, how can Egypt recover and maintain its position as the leading power in the eastern Mediterranean for centuries?
Recover from what? Egypt is in a sort of cosmopolitan golden age around the time of the Pyrrhic war and remains a center of art, learning, and culture for the next few centuries.
How much better can things get? In terms of political power, you'd need to weaken the Syrians and allow the Ptolemies to gain hegemony over the Mediterranean, which is certainly possible might best be done by butterflying the Seleucid dynasty (before your PoD) or having it succumb to a lot of regional rebellions (quite possible, just have more Satraps go the way of Baktria).
However you'll also need to avert the rise of Rome somehow, and with a PoD after the Pyrrhic war you're harder pressed to do that without a series of major divergences additional to whatever one broke down the Seleucids. Still, perfectly doable I expect, and you'll end up with a pre-eminent Lagid thalassocracy that could, with some reforms, probably last a few centuries. Of course you might want to find a way to get them to tone down the inbreeding as well, before that destroys them.