I won't lie, it is completely out there, but I've always wondered if there wasn't a Tartessos-Sea People link. After all Findley does suggest that there was a great number of migrations that triggered it all, I don't see why Spain wouldn't be included in that, and have its cultures impacted. Perhaps the Tartessos of the 8th Century BC is the result of migration into the region that displaced whosoever was there before.
I mean, it doesn't overly matter because of the cascade of different invasions, but it came to mind.
But as to overrunning Egypt? I'm convinced it is possible, but I could see them establishing themselves in the Delta and essentially cascading Egypt into another 'Intermediate Period' - which all in all might be a damn good thing. The overall decline of grain production, famines, etc that characterised the Bronze Age Collapse probably won't collapse all of Egypt, but mainly the central authority, leaving the various Nomarchs to fight amongst themselves.
One of these could well forgo chariot-style warfare with all its complex prerequisites for a more 'Desert Horse/Valley Hoplite' affair, and rebuild a new Central Authority with a revamped style of warfare, and decentralising some of the more centralised practices (i.e. letting farmers or locals determine what to grow, have their own seed stocks, etc)
THAT 'Iron Kingdom' Egypt could be stronger, more agile, and more dangerous than the Egypt that survived. It'd also likely be established around the Wadi Hammamat trade route (perhaps even in Quseir), as then they can import iron smelting from abroad, and exploit the mountains of the Eastern Desert in relative isolation before returning to the Nile.
What I don't see is the Sea Peoples going further. The Delta is a sweet prize. The Hyksos saw that, and if Egypt can reorganise and reunited after the Hyksos, they can do so again after the 'Sea Peoples'.