It may not have been issued on the basis of the EASA license, but it would have had to be updated to reflect the license number/format change.
Carrying both certificates would have been enough to legally fly the aircraft, “not for compensation or hire”.
The FAA interpretation of that is quite strict. However, enforcement most likely would have been up to other States, if he remained outside the US.
The other issue is the EASA license. With let's say a German license, the N-reg could be flown (for hire) in Germany, but not elsewhere in EASA territory. If I recall correctly, this is an FAA interpretation of EASA: Not one place, but a collective...
All too confusing, but I bet the insurers would have no problem figuring it out after an accident or incident!