As far as I know the three JAA pilot factories in Florida (OBA, OFT, Naples) don't do seaplane ratings so no chance of doing this on your JAA PPL in Florida.
So you either have to obtain a standalone or piggyback FAA PPL. Been discussed here over and over again. Should not be too hard to decide which one, and how to obtain it. (My $0.02: get a standalone once you intend to add ratings and stuff to it.)
Once you've got your FAA PPL in hand, piggyback or standalone, you can train for your seaplane rating. If that's less than 18 hours of practical work per week, you should be able to do this legally without an M-1 visa but you have to get the INS to confirm that in writing 'cause both the friendly INS official at the border and the TSA will more or less automatically assume that you have one, and will frown (at least) if you don't. If you do need or want a visa, you need a school that is SEVIS approved and not all flight schools in the US are. Part 61 schools are by definition, and part 141 schools are usually not. Or the other way around, I don't remember. But only schools that are SEVIS approved can "sponsor" your visa: are able to issue the paperwork necessary to obtain one.
As for TSA, I'm not sufficiently familiar with the FAA seaplane rating (is it a class rating, an endorsement, differences training or something else) to judge whether you need TSA clearance. It is commonly known that you need TSA for the initial PPL, for an ME and an IR, but I honestly have no idea about a seaplane rating. You might want to contact the TSA about this.
In any case, do you plan on exercising seaplane privileges once you're back in the UK? Otherwise you can just go to any seaplane-operating outfit and do a few seaplane tours with a CPL (doesn't even have to be an instructor). I'm sure the CPL will allow you to handle the controls after a while so you get the experience and the fun, but not the logbook or license entry. And you miss out on a whole lot of bureaucracy...