You have to contact the FAA, specifying an FAA District Office, and get your national aviation regulator to verify your licence to the FAA first. (Check FAA website). They will eventually send you a letter giving details of how to contact your specified FAA office (FSDO). You will have several months in which to visit the FSDO in person.
I rented a C152 from Fairbanks many years ago, with the same procedure as in Colorado, S.Dakota, Oklahoma, and Montana. Just a check flight with an instructor However there may be restraints on taking the aircraft to some areas. I was offered a survival kit if I wanted to do so.
Unfortunately, one of the FBO's C152s was going for an annual, and someone damaged another, so rental became restricted for the rest of my time there, and I flew less than I had hoped.