Anything is possible if cost is not an issue. Plenty of G-reg planes fly to the USA commercially so CAA licensed engineers must be out there.
Did Kirk buy an old N-reg and rapidly transferred it to G, while actually out there?
All ICAO flight training is allowable towards an FAA license, so all your Swedish PPL training is allowable. You will need to do a few hours extra in the USA (in particular some night time perhaps), do the FAA medical, written exam, and then the oral and checkride.
The resulting standalone FAA PPL will be yours for ever. To keep it current, you just need an FAA Class 3 medical and a BFR every 2 years.