Interesting plot if you have the time is to take an FAA rating whilst you are out there and then your licence becomes an "original issuance" which is less restrictive and also means you can fly N reg aircraft anywhere in the world.
I'm not sure whether your restricted FAA licence would allow you to go to Europe as pic of an N reg a/c.
Multi rating is v cheap compared to Europe and easily done in 2/3 days.
Just a thought.