Once you have your EASA ATPL verification, all you need to do is complete the ATP-CTP course and then the flight test.
You can do the flight test in a light twin (seneca) if you want. Or the biz jet type (in a sim) you already fly. It just has to be be multi engine complex. No minimum weight.
You do not need a student visa. You will need to get finger prints in USA for sim training.
I did this in 2018 with an ICAO ATPL.