It’s odd that Cathay make you proactively request an Invitation to an Arrivals Lounge thousands of miles away while still checking in at HKG. If that really is the case, then the procedure needs to be changed.
AA accepts both BA and QF Boarding Passes (as well as its own exhausted pax who have had to endure a wedgy bed and geriatric cabin crew on their flight over) as proof of purchase and grants access based on that piece of paper. Can Elin Wong confirm that AA will accept CX Boarding Passes as proof of entry at its LHR AArrivals Lounge?
This would best be done by calling the AArrivals Lounge LHR Desk direct to eliminate any risk of policy vs. practice deficit.
CX might also ask AA to update its LHR AArrivals Lounge information I linked in an earlier post in this thread. It is not very clear whether other eligible classes flown on oneworld partners are permitted access, and what is required to grant access (invitation or simply a Boarding Pass). This could be easily fixed.