Hi, I am a solicitor working at a firm in the City. In relation to what Freshfields have told you, while that is the line they peddle, in reality law firms are very, very interested in the university you went to. It won't form part of the "official" criteria upon which they select, but it will be certainly be something they look at. And frankly, a low-level LLB (even if you get top grades in your modules) would, in my view, not be looked on that favourably. If you don't believe me, take a look at the profiles of junior solicitors at City law firms including Freshfields - I doubt you'll find anyone who has attended a non-RG university.
I would try to do the best degree possible at the best university - you should then only do the GDL and LPC if a firm sponsors you. Of course, this route will mean you will incur some extra expenses in relation to the living expenses for the GDL (on the basis that a firm's sponsorship will not cover all of them), but it should be worth it.
I also have the NY bar - be aware that it is of ZERO help in getting a training contract. I did it for different reasons (I wanted to pursue a career outside of the UK). I happen to also have an LLM - again, it will be, unless it is in a niche area like competition law or similar, very little use in getting a training contract. By all means do both if you want to for whatever reasons you may have (I massively enjoyed doing both and they were enriching experiences), but please don't feel the need to do them to get a training contract.
I hope this is of some use. Ultimately however the decision is yours, but as a (junior) solicitor at a good City law firm, that's my view.