Cambridge generally isn’t as keen on Americans as Oxford (see my post here: http://talk.collegeconfidential.com/united-kingdom/2091170-oxbridge-admissions-for-americans.html#latest) so bear that in mind when choosing between them, though I assume you picked Cambridge because the combination with History or Management at Oxford doesn’t appeal? Have you tried the ECAA test for Cambridge https://www.undergraduate.study.cam.ac.uk/courses/economics#entry-requirements? Remember your school needs to register before October 15 to administer the test. And does your reference know what he/she has to write, will they predict all 5s on this year’s APs?
Generally I’d expect Cambridge and LSE will be hard to get an offer from, UCL somewhat easier but not a really high probability and Edinburgh and KCL more likely. Also I’d expect that any offers you get would be conditional on a 5 in Calc BC, so you should have some backup options lined up in the US unless you plan to take a year off.