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That's weird that the LHR regular Admiral's club claimed the OP did not have authorization to enter. I recently used my Alaska lounge card at the Admiral's club for an outbound AA flight at LHR in January. I had no issues getting in.

I always carry my Alaska lounge card with me. It's just easier when dealing with Admiral's Club lounges rather than trying to show them a phone app. Then, if there are any issues (like in London), you can show them your card and the listings on the Alaska website. If you are still denied, email Alaska and ask for lots and lots of miles as compensation (I did that when I used to had AA platinum status and was denied entry to the Finn Air lounge). I also carry my Alaska MVP card if traveling through Iceland so I can gain access to the very nice Saga lounge (most incredible food selection I've ever seen in a lounge). I can see though why entry would be denied to the AA arrivals lounge at LHR. Nevertheless, it is not clear on the Alaska website. I would email Alaska and ask for miles for denial of entry to the AA arrivals lounge and see what happens. The worst they can do is say no.