US to Europe is not a big deal over all. You will have issues with the pilot certification. Primarily under ICAO rules the nationality of the pilot certificates operating the aircraft must match the nationality of the Aircraft or the airspace. Since it is an "N" number aircraft you will either need FAA certificated pilots or they will need a validation. I have had FAA inspectors in the past who would issue a temporary certificate or an LOA for the flight. Check with Airman Certification Branch in Oklahoma City for your options.
If you deregister the aircraft prior to the flight, you will very likely lose all the RVSM authorizations as they are linked to the Registration.
Good luck and have a fun flight.