Congrats on getting into both programs!
This is what an admission officer who works for the UN and attended a UK school told me: if you want to work for the US State dept, or as a US foreign service officer, you should go with an American school. However, LSE is pretty much an all-around stellar school and in my opinion, more reputable than UCSD. I think that LSE will open more doors, especially if you're American, it will open the door for jobs outside the US.
Another thing to consider is to make sure the program offers the skills you want to learn. And I am talking strictly about skills (so stats, econometrics, writing, research, etc.), stuff you need on your resume to get the types of jobs you want.
LSE is very expensive in my opinion, but I guess for Americans, it's just a normal price, haha! Keep in mind London is one of the most expensive cities in the world. But like you said, it's just one year, compared to two in the US.
If I were you, I'd pick LSE (but I don't know that specific program, I am just basing this on the schools).