I imagine this could be done sometime during the late 18th or early 19th dynasty most easily but it's up to you. My challenge is to have a Kingdom that is doing realistically well (perhaps with less external or internal conflict than OTL and a more sea faring culture emergent) colonize and populate southern Europe to any significant degree - effecting its culture and genetic makeup longterm.
The Egyptians do not have to (and almost certainly could not) establish any sort of lasting imperial colony across the Med Sea, but preferably there would be enough of a presence to change the linguistic, cultural, musical, genetic, and technological course of the colonized area.
It also does not have to be a military or strictly military conquest. There could be a peaceful immigration of Egyptian peoples for religious, political, necessity, or even adventurous reasons.
Once the challenge is fulfilled, I'd like to discuss the effects of a modicum of Egyptian-ness in early Greco-Roman development (say Egyptians conquer or infiltrate Greece just as the Mycenaeans collapse, and infiltrate southern Italy around this time as well establishing trade routes around the Mediterranean) into the coming centuries.
Once the empire collapses in Egypt, will Greece and Rome still develop along similar lines, but with a tinge of Egyptian-ness leftover? (I hope so)
What do you think about a semi-Egyptian southern Europe?