David should have known that Rose's real name was Sarah long before she told him, since they're neighbors and he sometimes receives her mail by mistake (unless she gets mail under her stripper name, which is unlikely).
It might've been addressed to Rose's boyfriend.
He also bought her an airplane ticket... without knowing her name? How that's possible?
Unless they used the fake papers with her fake Miller name on the plane. The papers they would have to have in order to pretend they're a family on the way back.