It seems things will be getting easier for FAA to EASA licence conversion thanks to the recent Bilateral Agreement between both authorities. I think this is a route very worth exploring. Go to the USA and get an FAA licence, then come back to Europe and convert to EASA.
If doing so, I would recommend OP check very very carefully what the different flight logging rules are in each authority as there are some important differences. It might be best to record each flight with all the details required by BOTH FAA and EASA, keep an Excel spreadsheet to make things easier. Or two logbooks, one for each authority.