Deport everyone from Gaza to Yemen.
Deport everyone from the West Bank to Syria.
Piquant, with much merit. But why not Amman? That's where they all come from (plus some Egyptians). The Jordanians threw them out to begin with, so they won't be happy to see them back—to the point where they'd probably shoot them on sight. But it won't involve Israeli or American personnel—or waste our bullets.