There's a digital version of your BAEC card in the BA App. To me, that is a more logical place to look (plus of course is consistent across different platforms).
Company apps are fine but they can be slow to open and you normally need a connection to access any details. Otherwise, why do airlines support Apple Wallet for boarding passess if their app does the job?
I would love more European businesses to embrace Apple Wallet (I could easily lose KLM, BA, Nectar and Tesco cards from my wallet). Costa seems the best compromise, they have an app but if you just want to store the card details in Apple Wallet, you can just do that as well. Stocard is OK but doesn't work in all situations.