So just go and get an FAA medical then fly as much as you like in the states on a 61.75 conversion.
You can get a sport pilot certificate in a few hours - you can fly up to 1320lbs with no medical - (just a US driver's licence can be got in a few hours) so a Piper cub would be ok.
Or can get a recreational certificate - and fly the usual 4 seaters (only one passenger day VFR) - with no medical.
Flights and accommodation will cost about £500.