Timpson can defintely do the 1st gen keys/fobs, but any good locksmith in USA should be able to do this at a reasonable price and more convenient for you in USA.
Cutting the key is no problem, but normally you would test the key/fob with the car before you leave and any problems can be rectify easily.
The new key fob would need a pairing process with the car which you can do yourself.
Check out or any other locations with "Remote Car Keys" services. https://www.timpson.co.uk/stores/bath-stall-street/
Not sure if you're on "North American Motoring", but if you google and post in that forum you might get someone who will know some locksmith in USA closer to your location.