All Correct, but.....
If you do the written exams with FSI you'll have to sit the tests at Paris. It's probably worth giving Melanie Stewart at Soloflight a call to see if you can do them there. (I did mine there and the service was excellent)
I approached the problem from the same angle as you and found that reading the ASA test prep guides through gave me more than sufficient knowledge on top of waht I had learned from the JAA groundschool to do very well in the tests.
With regard to the TSA, its not simply a matter of turning up and having your fingerprints taken. you must follow the procedure:
1. Contact a training provider
2. Create an Account at
https://www.flightschoolcandidates.gov
3. Create a Training Request having ascertained what catagory you'll be applying under.
You'll then be invited to pay a fee for the Training Request (about $130 I think) after they have confirmed knowledge of the request with the Training Provider concerned.
You'll also need a visa to train in the USA. As with the TSA process I'd liase with the Training Provider on what the latest advice on the process is. Do however make sure that all the paperwork is in order AND THE DATES MATCH before attending your visa interview at the Embassy.
With regard to training providers, it would depend on your objectives but check your PM's for some recommendations.
I hope this is of assistance.
HS125