Transiting through LHR for LHR-PHX (on BA metal) PHX-MEX itinerary all booked on 1 PNR through AA.com

I already had my BP for the LHR-PHX leg so thought I would save some time by getting the BP for the PHX-MEX leg at the transfer desk at LHR. AAgent said they couldn't do this as my tatl was on BA (made a big deal of saying "in the old days we could")

Thought I'd try the physical equivalent of HUACA so went into the lounge at LHR and was told no problem here you go

Who was right in this case?

Additionally - waiting at PHX and looking to get a BP for tomorrow's MEX-LAX flight and have been told they can't do it and to check in in MEX - are they able to do this or do I really need to check in manually?