If you are planning to be there a few weeks then you may likely require a M1 visa. The school may demand that you have the visa for their own protection.
I doubt it. First of all you are not training for a FAA certificate or rating and besides, it is a
renewal for a rating that you already hold.
Recurrency training, flight reviews, conversion training, check out for rental etc etc are all exempt from the M1 visa.
I know of a UK CAA examiner in Vero Beach, Florida.
He can be reached under the username "keygrip" on this forum.
Regards,