It can all be done in Europe.
The training is easy. The FAA accepts all ICAO foreign training so your existing logbook entries all count towards any FAA paper.
The medical is easy. Plenty of FAA doctors in Europe.
It is the checkride (the flight test) which people have had a lot of "fun" with in recent years.
If you already have a JAA PPL (you say you already have a plane) with the night qualification then I suggest doing the written exam here, the medical here, and then pop over to the USA for a few days (allow a week) and do all the flying and checkride out there.