Why not both ?
If you are going to spend some time in the States I would be tempted to just get a Stand Alone FAA Certificate. The exams are fairly easy and rather than just punching holes in the Sky you will get some different flavour flight instruction. Having a US licence may be of use in the future.
As regards BFRs I do them in the UK with an instructor who has both JAA and FAA privileges, do the 1h flight and he signs it off as meeting both Authorities requirements.