It just isn't necessary to register with the TSA for renting an aircraft and a BFR is not training, it's a flight review (biennial flight review) and it's not considered training. It's pointless to do the BFR in the UK because when you rent an aircraft. You will be required to demonstrate competence both on the ground and in the air and that is essentially the same as a BFR so it's pointless paying to jump through the same hoops twice.
Some folks are taking this TSR stuff a bit too seriously. Essentially, if you aren't going to do actual flight training, ie PPL, CPL, IR then there are no worries. I promise!
Just go and enjoy.