Look for threads on "piggyback" FAA PPL on here and Google generally.
There is a relatively straightforward procedue to be issued an FAA PPL, if you plan to rent from a school or FBO used to renting to UK pilots they will be sble to take you through the process.
Also the TSA clearsnce procedures.
Be wary of deals that seem too cheap - even block booking $83 sounds low $100 -115 sounds more realistic, plus the renter insurance spread over the hours - no matter ehat it'll still be cheaper than the UK.