Stuart,
I was in the same position as you. I was getting my FAA PPL, based on my JAR PPL. Like you I assumed that as long as my JAR one was current, I could use my FAA one, which was based on it.
But to be sure I asked the guy in the FSDO, and he made it very clear, that I must have a BFR before being able to use my FAA PPL, and every two years there after. This is in addition to keeping my JAR one current.
dp