Stout you have a PM, with a checklist for US training.
The easy mistake to make is to book with a Part 61 school; the TSA website does not warn that you need a Part 141 school to issue an I-20 which you need to get the Visa. I wasted a load of money with the TSA too, doing this.
There is a recent development, I believe, in that the TSA now advises the Immigration of TSA applications, and all of them now need a Visa. This is despite US LAW not requiring a Visa for training of < 18hrs/week. Maybe somebody can confirm this.
This kind of crap, plus the crude interrogation one gets by the Immigration "staff" when going to the USA, not only for the training trip itself but potentially for years afterwards when they see you once held a Visa, is why people spend £ thousands and thousands trying to do FAA checkrides here in Europe.... then you have to comply only with TSA stuff which is basically just fingerprints and scanning the passport. And TSA is not required for the CPL.
All ICAO training, in any reg aircraft, is allowed towards any FAA license/rating (not many people know that). Only the last few hours needs to be done with an FAA CFI/CFII, and of course the checkride needs to be done with an FAA examiner.