I think we disagree as to the relative success rate of LSE in getting students into MC law firms, but 1) the OP wants to get an offer, and there are other unis on the list where the odds are clearly better and 2) the OP’s odds of getting into LSE are (imo, obvs) lower than some of the other unis on the original list, making it still the one I would switch out first. But, as they say, difference of opinion is what makes a horse race!
More importantly, I think that it’s a very problematic plan for somebody who sees top tier US law schools as out of reach. The students who MC firms will work to get visas for are top-tier students, who will have top-tier backgrounds (usually, but not always top-tier unis).
OP, what do you think of Exeter in the autumn as an option?