Hi!
I studied anthropology at LSE as an undergraduate and had contact with a lot of postgraduate students as we attended lectures for optional courses together. Quite a few of them were from the US!
As a US citizen unfortunately you are not eligible for a UK state loan - soon enough even we EU students may not be , but it seems you may be eligible for a US loan to study at LSE - http://www.lse.ac.uk/intranet/LSEServices/financeDivision/feesAndStudentFinance/feesAndLoans/LoansUSA.aspx.
The people who will handle your application at LSE really care more about your fit for the program than grades, assuming you meet the minimum. I would concentrate on writting an excellent personal statement in order to become competitive Hope this helps.