So to summarise then,
1. You have a FAA Temporary Airman Certificate, not an FAA PPL.
2. To fly under PPL privileges in the USA, you need to have this certificate plus a valid UK licence.
3. To have a valid UK licence:
a. Your licence must be within the correct validity period.
b. You must have a valid rating (e.g. SEP Class Rating).
c. You must have a current JAR medical certificate.
4. You do not need an FAA BFR to keep your temporary airman certificate valid.
.....is that correct, Noggin?