Like others have said, the reputation difference is negligible. No employer will care if you went to Bristol over Kings or vice versa; they’re both very good Law schools and a degree from either looks excellent.
You’re on the right lines thinking about whether being in London or outside suits you; I personally wouldn’t want to be in London for an undergrad, but it might suit you perfectly. Wouldn’t even say London has a better nightlife than Bristol — there’s obviously a lot more choice, but it’s much more expensive and realistically you find the 1 or 2 clubs you like and stick to them so don’t need loads.
I’d also take one of your choices out and have a “safe” one in there. Oxford is obviously very hard to get into, and the others you’ve picked are amongst the hardest to get into in the country; LSE, UCL and Kings have a 20%, 19% and 23% offer rate respectively. Durham and Bristol are easier to get into, but still difficult.
It’s good to have a “safe” option that you can bank on getting an offer from regardless and is still a very good uni. In the instance your LNAT goes horribly, you could be left with nothing. You mentioned Exeter — that’s a very good choice. Lovely uni and campus, high offer rate, good links to firms, great course, and honestly no worse than Bristol or Durham.
If you go for Oxford, LSE, UCL, Bristol and Exeter say, you’ve got a better range and one safe non-LNAT choice. Hope that helps.