Do I need some sort of FBO or Training Certificate for the club in order to allow me teach students to the FAA sylabus in their aircraft?
There are two training routes to an FAA rating, via
Part 61 of the FARS (or 14 CFR) and
Part 141.
For part 61, there is no syllabus as such. You turn up and receive instruction as the CFI sees fit. For part 141 training there is a syllabus, and you have to follow it. The FBO has to be approved by the FAA to provide part 141 training, and if you go to a Part 141 approved training establishment they will have a certificate on the wall saying so.
You take the same exam/oral/checkride for both Part 61 and 141, and the certificate is the same. Being more structured, Part 141 provides a quicker route to the rating, though in practice there is not much difference at PPL level.
So do I get the impression you are looking at teaching FAA students in the UK to build hours? Ask yourself..
Why would anybody want to get an FAA rating in the UK when it's so much easier, cheaper and relevant to do this in the States? How are you going to sign the student off to solo in UK airspace when you don't have a UK authority (instructor's ticket) to do so? Where are you going to find an FAA examiner in the UK? What do you think is going to happen with this new security and background checking move? It wouldn't surprise me if all training for FAA ratings will soon have to be provided in the US by approved establishments.
Good luck