Batty
BA looking was nothing to do with modular/integrated - it was all contacts, which the integrated schools had through their sponsored students, but they were broadening into other schools. I know this because they were also considering students from SFT which only ran modular courses. I know one that would probably have been offered a place (then would have lost it in September) had she passed her IR first try.
no sponsor
Considering your current hours will not cut the integrated course, you will be paying a premium of around £20,000 for the integrated course over the modular. Many modular courses are based in the same classroom environment as integrated, often taught by exactly the same instructors. Since all are approved the instruction is in any case of equal quality.
If all else fails (as it can on either route) £20,000 will buy an awful lot of private tuition.
The time taken on a modular course need not be any longer than an integrated.