No one should have to 'prove' they're exempt i.e. disclose personal medical information to a stranger.
However, that woman was belligerent and antagonistic, and that's why she got removed. You think the police were called simply because she didn't have a mask on?
The police would never have been involved if she hadn't been flapping around mouthing off like that, and I bet she was worse before the uniforms turned up.
I'm exempt, I can just about manage going in for one item and self checkout at quiet times. Other than that I haven't been anywhere. If however I had to do a proper shop, I'd approach security and just explain I'm exempt and ask what to do, politely, quietly. They may have a system in place to run round and get my few bits or possibly a delivery slot for these situations I don't know about. I have a lanyard but they're worthless these days, but it's the attitude that staff will notice.
Don't fucking go striding in with an air of 'you can't tell me what to do! Ask me why I'm not wearing a mask, go on, I dare ya!' and then cause a scene when asked. If you're causing a scene to the point the police are called, YOU are the problem.