I'm not a politics student, so I realise I only have very limited relevant experience - but if I were you, I'd choose the course that appeals to you the most.
I'm a current master's student at LSE, and I can assure you that not everyone is status obsessed; I've met a lot of fantastic, down-to-earth people here, and some truly inspirational professors - but that being said, I'm not a huge fan of our campus or the university's general 'vibe', which sometimes makes me feel more like a customer than a student. From what I've heard KCL has more of a traditional university feel - but to be honest, I don't feel like the fact that I don't like my campus matters that much; I'm only here for a year, I'm too busy with work and friends to get very involved in student life, and it's London, so there's always better things to do than hang around campus, imho!
As for LSE's prestige, I think that how relevant that is depends on what you plan on doing next; I'm an EU student, and hardly anyone in my home country has heard of LSE, so if I were planning to enter the local job market, it would hardly matter. If you're planning to work in the UK, I guess LSE might have a bit of an edge over KCL (depending on the subject - not sure how they both rank for politics), but KCL is a highly respected university too, and employers would likely be more interested in what relevant skills you'd acquired than just how highly your university was rated by the latest QS ranking.
I'm happy to answer any questions you have about LSE - but I think that, between LSE and KCL, there really are no 'wrong' choices, so just go for whatever course you prefer!