I know that for an american student to be submitted to LSE its all about the AP tests scores, but do they look at your high school transcript or any other sort of indication of academic performance
Short answer, no.
They can see your grades (in that it is on your UCAS form) but there is no evidence that they pay any attention to it. There is no specific SAT number (all they say is that the SAT is not sufficient), but it’s safe to say that you would probably be better off having an SAT over 2000.
The personal statement should be pretty intellectual / thoughtful / incisive / etc. and (indirectly) make it clear that you know what you are doing. Make sure that any major achievements are in either your PS or your recommendation.
If you really want LSE and are not applying to Oxbridge, send in your application a little after 15 October. There is a view that LSE is a bit competitive with Oxbridge, and I know several people whose conditions for LSE were harder than their Oxbridge conditions- and everybody knows that 15 October is the Oxbridge application deadline. Completely anecdotal, but it can’t hurt!