It's not a BFR. It's a flight review. It's not biennial...
To be PIC you must have had a flight review under FAR 61.56, or equivalent credit.
"Conversion" does not require a current flight review if done under 61.75. That's the paperwork part. To actually fly the airplane requires 61.56 compliance. This could likely be done as part of a rental checkout - something that's going to be required anyway.
ICAO is not a country. If you have a license from an EASA nation you can use the new bilateral agreement to get an FAA certificate that is independent of the foreign one (unlike 61.75). This path requires a current flight review as part of the issuing process. And a written test. But gives you a "real" FAA certificate rather than a piggy back on.
Canada to US is easy. EASA to US is almost as easy. Anything else is either 1) 61.75 or 2) just get a real certificate by taking a written test, doing some prep/training, and passing a checkride (the checkride gives you 61.56 credit).