I would strongly disagree with some of the responses you have received. LSE is the most prestigious of these but all three are great universities and have the best IR programmes in the UK. LSE is the most prestigious of the unis but IR is the flagship course at St Andrews and arguably at King’s (alongside war studies). So all three are great options.
LSE might give you a slight advantage but like you said, there’s no point going to LSE and having a terrible time and then getting a bad degree when you could have gone to St Andrews and had an amazing time and got a first.
It really depends on what you’re most interested. St Andrews focuses a lot more on the Middle East whilst LSE is more theoretical in its approach with IR as a discipline as opposed to specific regions or events. I’m not sure about King’s but I imagine foreign policy and war is a big component.
Socially, again it depends on what you want. If you want a small community feel where everybody knows each other and there’s loads of balls and events then pick St Andrews. It has a great pub culture and is very sociable. However, if you’re easily bored and enjoy bigger cities - don’t go there! It’s a very small place with no train station which is part of the charm but isn’t for everybody. Obviously both LSE and King’s are in London and naturally, the social life isn’t as great. This is the same for most London unis; higher proportion of international students which is a great thing but sometimes they stick in groups and halls are generally less sociable. You need to make more of an effort but having a big group of friends is still possible if you join lots of societies and put yourself out there. Internships are much easier as you’re in London. King’s definitely has a better social reputation than LSE as does UCL. LSE like you said is known for being extremely studious and lacking a social life and it doesn’t have much of a campus which contributes to that. However, something you can do is make sure you’re in the more sociable LSE halls like Bankside which can help. Or, as an LSE student you can go to University of London halls (Garden Halls and Connaught Hall are supposed to be really nice and sociable) where you would be with LSE students but mostly UCL students and you are located pretty close to LSE but live very close to UCL’s big socially active campus. LSE also has more work which is manageable, you’ve got the grades to get in but if you’re looking for a healthier balance then maybe St Andrews is a better option for you (keep in mind at St Andrews you would do 3 subjects in your first two years and then just IR for your final two) - is that something which appeals to you? St Andrews will feel closer to a school environment whilst LSE feels more like a work/job one. You also have to take a compulsory social science module in first year at LSE and yes it’s true there are more postgraduates at LSE than undergrads and I’ve known a lot of people who have gone to UCL, Durham, St Andrews etc and had a great time at undergrad before doing a masters at LSE.
It’s really up to you with what you decide to do but don’t feel like you have to go to LSE. Sure, LSE is seen as the best of these three but St Andrews isn’t far behind at all and IR is its most prestigious course. King’s is definitely weaker than both in terms of overall reputation but again, it has a great reputation for IR.
Another thing to consider tho is where you would like to work. If you want to work in the UK, I don’t think going to any of the other two would put you at a disadvantage compared to LSE at all. However, LSE’s international brand is stronger than its national brand if that makes sense. Here in the UK, it’s not seen as that much better than say UCL or St Andrews or Durham but outside the UK, LSE is certainly the best known of the UK unis besides Oxbridge. King’s will also have a decent international brand and be known but not to the extent of LSE. St Andrews has the weakest international brand name but it’s improving and it’s arguably just as known in the US as LSE these days.
But please don’t feel like you have to go to LSE, it’s an amazing uni and you did the hard part - get in, but the other two are excellent choices and getting a first or a 2:1 from them would be better than struggling and being miserable at LSE. But there are ways to improve your social life at LSE so have a think about the courses specifically.