Correct me if I'm wrong anyone (I'm sure you will) but if you have a PPL already, any further training is modular by definition. Integrated courses just slot all the training together and candidates can leave applying for licences etc til the end of the whole course since integrated training had dispensation from some of the 'you must have x before commencing' type parts of the syllabus.
All FTOs doing training of the sort you want are 'approved', in that they all comply with CAA's CAP 682. Approved in the BA case merely means that BA has a preferred supplier arrangement. At present they seem to like Oxford but that could of course be altered in the future. Broadly speaking, the schools most often mentioned in this area are Cabair, Oxford and possibly CTC (though they facilitate triaining rather than owning a bunch of aircraft). There are several others who offer integrated courses - a search on this forum will reveal loads of info...
If you're thinking about integrated as a way of gaining an advantage in the job market, once again a search will reveal what appears to be a raging debate on this subject! Either way, best of luck.