Am I alone in just thinking this FAA.v.JAA issue is a daft question.
FAA and JAA requirements for that instructional flight are not the same, and there is no reason to expect an FAA instructor to be aware of JAA requirements (even though they are pretty minimal) or a JAA instructor of FAA requirements (which are rather less minimal).
Holding a JAA licence I'm clear than my biennial flight has to be with a JAR FI or CRI.
Holding a FAA licence I'm clear that my biennial flight has to be with an FAA CFI or CFII.
I can't honestly say that either of these things gives me a problem, nor with the fact that they are different flights (okay, if I could find somebody who is both an FI and a CFI then they could I'm sure be combined - and a few such do exist.)
It really is just how it is so far as I can see. Live with it!
G