The CAA is the legal authority, the FTO (school) is the training provider.
The FTO prepare you for the exams. The CAA host, mark and validate the results of your exams. Once you have sat and passed your ATPL exams, they are done and the results are held by the CAA. You can't redo them; because you don't need to!
Whether or not you can join another integrated course is nothing to do with the CAA; they are there to facilitate the issuance of licences and ratings to ICAO/national standards.
To answer your question, it's down to the relevant school and their policy, though if you have already completed your ATPLs I'd imagine most FTOs would want to steer you in the direction of the modular route.
I hope this makes sense.