Just to be clear, a BFR (strictly speaking officially now just called a 'flight review' but still biennial nonetheless) is the legal requirement to maintain an FAA private pilot's certificate. I do not believe it is necessary to have one to validate a new FAA certificate issued on the back of a foreign licence, although I may be wrong. If anyone knows differently I would be interested to see a reference.
A checkout has no relevance to a pilot's certification. It is simply something that any renter will require before letting you disappear with their $100000 machine.