But if you do not have a JAA licence - you can fly both on the FAA?
Then the FAA licence is not "Restricted" and you can do what ever the FAA one permits, the JAA one is irrelevant.
If the FAA Certificate is "Restricted" it is only as good as the privileges on the one it refers to, so if that is not valid then neither are valid.
You can only render valid in one State what is valid in another!