IF you can get a BP for LHR-LAX from BA then that gets you into the FCC with no hassle.
Once you are in the FCC you can walk down to T1/T2 or get a bus to T3 or T4 without being challenged. If you need to get back from T3 to T1I then you can do that again fine through the FCC.
You don't actualy need a BP to get into the FCC, just some form of confirmation that you are on a flight that day. So presumably a printout confirming you are listed for standby would work as well if you could get hold of one.