I know DUB can do AUP on the DUB-LHR service, so in theory they can do the same for LAX. It may be the agent you have in front of you won't do it or is reluctant. If you have time to get to Flounge then in theory you have time to do it in LHR. But assuming you are going via the First Wing (as you should since you don't need to go via the UK Border) then it won't take a second to ask there.