Sendsummer, in practice the more expensive halls tend to attract overseas students. The older, cheaper and often closer halls (Passfield - which was quite run down when I was there - though I assume they have done something since, Roseberry, and Carr Saunders) are more popular with home students. The issue, in London, is that you have to make do with less. Just as highly paid bankers live in ordinary terraced houses in Fulham, London students don't expect to have an ensuite, or a living room in their shared flat. Several of the intercollegiate halls are also within walking distance.
LSE guarantees accommodation to ALL first year students. From memory you get five choices. So easy. You pick the three LSE halls mentioned and then a couple of others, perhaps including a couple of nearby Intercollegiate Halls. My guess, on the basis that DS did not hear until August, is that non London students are given priority.
The really rich students live in Mayfair, or in the expensive halls and commute in by taxi!
DS was in International Hall, which has a fantastic location but is huge. And in fairness he made little to no effort. However, as I said, plenty of social life happens on campus and he was active in three or four, and made friends with a group of second years who lived in a nearby flat, who will be his University friends for life, as will be the course friends he got close to in his third and Masters years, several of whom, like him, are now pursuing PhDs in the US.
Outside of lockdown there is so much happening in London, lots of which is cheap or free. So no reason why you should not go with friends from the music society to concerts, or hear speakers at different colleges. In his first year DS joined a school friend at Imperial in one of their non competitive sports societies. He was a member of a casual 7 a side football league which involved students from several central London University, and as part of an LSE computer gaming team that was regularly trounced by Imperial and UCL. I've met London students working in high end restaurants, John Lewis, and, a self-confessed foodie, in Borough Market.
London has layers, and student London is a very different place, to the London workers and commuters know. It offers some very real advantages, and some obvious disadvantages. It depends on what you are looking for. I suspect also you need to be a self starter and engage: on your course, with your fellow students from a diversity of backgrounds, in the University, and with London generally.