If it helps, LSE is notorious for relying on the PS more than most other universities because, as someone above pointed out, there are so few other data points for it to consider (no interviews, no LNAT).
You already know that your PS wasn't ideal, let it go, no use worrying about it now. You have great grades and a strong background, I'm sure Warwick and KCL will take that into account assuming you did decently on the LNAT as well.
As for Cambridge, as long as you make it to interview, that's all that matters. Once you make it to that stage, it's about killing that interview and everything else, to some extent, falls to the wayside. Onwards and upwards.
As someone who did do the LLB at the LSE, you wouldn't be missing out on much, the course is much like any other LLB course at any other good UK university. Your university experience is what you make of it, regardless of where you end up going.