Need to get a European medical too it seems.
Or just an FAA Class 2? A Class 2 medical is the same as Class 3 (PPL) except it includes an eye test, and it says "Class 2" on the piece of paper. And it expires after one year, though it remains valid as a Class 3 for as long as the equivalent Class 3 would. So just ask your AME to make it Class 2 instead of Class 3.
Last time I flew solo in the UK, you just needed to posess an FAA license and medical. Looks like they have decided to make it a lot more bureaucratic, with an explicit validation. France did the same thing a few years ago too. You used to be able to show up at the FSDO-equivalent (DGAC) with your licence and medical and they would give you a piece of paper which was essentially a French PPL. Now it takes weeks. France doesn't ask for a Class 2 but does ask for a medical issued within the last 12 months.