I cannot get a JAA 1st class but have been flying commercial in the EU for many years now with the same problem you have
Not sure you'd call 1 year "many".
may I ask how you are able to fly in the EU commercially without an unrestricted JAA Class 1 medical?
The answer is simple; don't fly a JAA-registered aircraft. One can fly around quite happily in an N-registered aircraft on an FAA license with FAA medical. That's what most of the bizjet operators do. I believe the Isle of Man also validate any ICAO-compliant license, so M-registered aircraft might well be maintained under EASA but can be flown with FAA licenses.