WrongWayAround,
The Regionals are all dropping their minimums and this is creating a real problem for the flight academies in retaining their CFIs. I would recommend that you talk to a few of the academies or flight schools and ask if they would sponsor you for a green card in return for 2 or 3 years of service. Check it out. It can't hurt and it sure beats trying to put on an act for 3 years. The INS has seen it all, heard it all, been there, done that, and got more than its share of t-shirts. Unless you really know the girl and trust her I wouldn't do that because if you're busted it means bye-bye USA for up to 10 years and with it any chance of getting into the US airline industry.
Good luck
MN