There's another thing you can do, perhaps the best of both worlds:
Get yourself the FAA license first. This will include the NQ. Then, after some 150 hours PIC (or TT? not sure at the moment, check LASORS, there is also a minimum x-country element) you can convert said FAA license to a JAR license by doing a very much reduced set of tests, obviously including a flight test. Voila! You now have two stand alone licenses. Only downside is, to maintain both you need to fulfill the med and BFR requirements for both.
On the issue of an FAA instructor giving a BFR in a JAR a/c. I looked into this once (albeit not in the UK) and the answer was that a JAR CPL was good enough. He/she may not even need this if not paid for the BFR, as under FAA rules the student is PIC (same, btw, during you initial checkride).
HTH.