What they said...…..just to clarify, you may legally fly with your FAA CPL and a Class 3 medical, as long as you limit your operations to PPL level privileges, so no flying for hire or compensation, i.e., "private" flying only. Not a big deal, plenty of pilots with advanced FAA licenses (CPL, even ATP), say, a retired airline pilot for example, flying (private privileges only) with a Class 3 medical.
Edit: oops, I just realized, this would hold true for flying in the U.S., but, perhaps the OP means to fly his N-registered plane in Europe? In that case, I don't have any idea how EASA and FAA rules fit together!