TSA and Visa's
I agree the need for a visa is based on the primary reason for your visit to the US. I did not bother with it last time I went to the US on business.
Particularly as the expense and hassle of an 800 mile round trip and a day lost for the visa interview did not appeal.
From experience the TSA don't ask for any visa details but they do want to know who you are from your passport.
They can cancel the training permission if they find out that you are there without the necessary visa - but then you would have to argue necessary etc. as discussed above.