So for non AAA members eg AA in the UK, it looks like you have to book via the non Marriott version of the website as those sites don’t request the AAA number. The app requests the number too.

Before the app and site were updated to force users to enter their AAA membership number I always used to book the AAA rate and never had to show my UK AA card. Travel websites also promoted the benefit of AA UK being part of ArcEurope allowing the discounted rate.

Unless I’m missing something I’m not sure what showing the Show the Card would do when asked for proof of AAA membership for non US guests.