The TSA rules are not too arduous. We use the interpretation from:
www.aopa.org/tsa_rule/index.html
We insist that our students have an M1 visa before they arrive here in the USA. I think there would be a problem if you completed the TSA application online without showing visa information. My advice is not to lie to the Immigration officer when you enter on a waiver and subsequently decide to take flying lessons. Any school that suggests that course of action is acceptable is plain wrong. If you breach your entry conditions you will be excluded and won't get back.
Regards,
Chopperpilot 47