As this question isn't specific to AA (because you can't access the lounge based on your class of service) I'd suggest visiting the
https://www.flyertalk.com/forum/airport-lounges-763/ Airport Lounges forum, which has tons of information about all the lounges at LHR.
Last time I was in T3 the digital sign for the Qantas lounge announced that it was "accepting paying guests". I had access via OW status so I don't know whether you had to be on a Qantas flight to get paid access or how often they accept paying guests (it was really empty at the time). However, there is a No1 lounge with pay access in T3, which is perfectly adequate for an hour or so (showers, etc). There's also a plaza premium arrivals lounge that you can pay to get into if you want to take a shower on your way into LHR.
The Cathay and Qantas lounges are nice but honestly, in the end they're just airport lounges. If you got such a great deal on your ticket, use the savings to treat yourself to a nice dinner in London, which will be better than any lounge food.