If your aircraft is N -reg no one can stop you flying it with an faa ticket except the faa ! The problem at the moment is that if you take you EASA ticket to the states they will issue you a permit to based on that license if you come the other way they won't hence the problem .
An faa license does not allow you to jump from a r22 into a 109 , you have to do training as required but there is no type rating and the training is with an instructor with no test at the end could be 1 hour could be 50 depends on the instructor who signs your logbook at the end !
Sorry about spelling grammar etc in a rush have to go flying !