All advice and suggestions are welcome.
You will need to go through a simple, but time-consuming process to have your EASA private pilot license "validated" to get an FAA Private Pilot Certificate issued "on the basis of" your current license. Lots of steps, but not hard. Google FAA validate foreign license and click on the EASA link on that page.
Note - there are no pilots licenses in the US. So, no CPL. The proper title for a piece of paper from the FAA is a Certificate; a private pilot has a Private Pilot Certificate (likely) with an Airplane Single Engine Land (ASEL) rating. ASEL is SEP in EASA land.
You will need to get a visa for flight training which, I believe, is different from a visa to live/work in the US. After you pass the FAA knowledge exam and the checkride, you will have a commercial pilot certificate independent of any/all of your EASA documents.