I’ve actually studied at the London School of Economics and Political Science! I did a law programme, which has specialisms which share courses with the MSc in Human rights. However, I am a Brit not an American so may not be the best suited to ask your questions but I’ll share my perspective.
Firstly, the teachers are amazing. There are so many famous and influential academics in the department. People I had been writing about during my entire undergrad were suddenly the ones who I were having a chat with in their office.
As to alumni networks. In the UK, for social sciences LSE is seen as a peer to Oxbridge (if not better) by people in the know - most international rankings show this. In South America, Asia and Europe LSE is extremely well respected. As such, it boasts an impressive alumni network - I believe over 50 world leaders studied here. In North America it is a slightly different story - I’ve found many people will have barely heard of Oxford. However, that being said LSE is an extremely international school and 10% of its intake a year are Americans! A couple Canadian leaders have studied here. JFK was supposed to study here but had to leave due to health complications. Jon Ossoff, who recently won the Georgia race, also studied here. David Rockerfeller, George Soros. There are many more of course. My point is that LSE has a great network despite it’s very small size and I don’t think you will suffer from an education at LSE. Any HR department worth their salt will at least have heard of Oxbridge and the University of London colleges (LSE and UCL). Moreover, I also hear that there are regular LSE alum meet ups in the states in the major cities (though I cannot vouch for this).
I’ll also add that LSE has a habit of sending people to NGOs and international bodies such as the WHO and the UN. Among others.
Furthermore, I don’t think LSE will hinder you getting into law school at Georgetown. LSE is well respected by US schools. It has formal academic partnerships with Columbia Law School, NYU and Berkley. During my time I met people from the Ivies and people who went onto top US school after their time here. So I wouldn’t worry about that aspect. LSE ranks 6th (according to QS) in the world for law and 2nd in social sciences (what it specialise in). Rankings are not be all and end all though, London is an amazing multicultural city. I’m still finding new things to do and I’ve been a Londoner my entire life!
One thing to consider is the price of London. It’s exceptionally expensive to live here. Also perhaps consider the likely situation of COVID in the UK around the time the course starts. The UK is currently in its 3rd multi-month lockdown.
Best of luck!