Keygrip, true...
But the school would be in the wrong should it commence training without the relevant paperwork and it courd all turn nasty should the the FAA, TSA, SEVIS or Immigration decide they want to check student paperwork. I would also imagine the likes of NSA and other bodies know who is going to flying school. PLus i am sure the JAA would certainly ask questions should the !!!!! hit the fan.
Bets bet is to speak to the FTO and see what they say. As far as i know flight training in the US requires an M1 unless it is a type rating under the offer of employment in which case a J class VISA will cover it.