Going through LHR on the LAX flights really isn't anywhere near as bad as going through on the bulk of the BOS, ORD and JFK flights most people have the bad experiences with.

FCC queues are worst in the mornings, from opening time until around 10 or 11. AA136 arrives at 1225, and AA134 arrives at 1450, meaning that FCC queues are normally negligable.

I'd definitly go for the one stop through LHR. Take the 134 on the outbound, enjoy the 2.5 hour layover in the Terraces with a shower and some lunch, and arrive in MUC in the evening far less stressed out than taking 3 or 4 flights and taking 14 hours rather than 24 hours.