Just looking again at the inside photo, which appears to show a flap of material over the badge area. Is this a repair, or a covering against the badge attachments? If the latter I wondered if the actual screw threads line up? If the flap doesnt cover them, it raises a question as to why its there, and if in fact it has the correct badge on at all. As such, it might lean more towards the theory its a foreign Air Force cap.