Just going to throw this out there for you as a student of human behavior and practician of observation with a masters in bullshitting acquired in a career of people lying to me...
Your answer, makes the answer obvious. It's also likely to result in further resistance and inquisition or hostility.
They WANT to hear that you're vaxxed, so tell them what they want to hear. "Are you vaxxed?" Yup! Just got my booster and I'm feeling grrrrrrrreat
!
If they ask for proof that's when you can say, hey it's not my appointment it's hers so I didn't bring my card, lost it, dog ate it, not required to produce my private medical records, etc.
Then it seems much less confrontational and obvious. I have to test regularly for work. Every doctors office asks if you've been tested in the last two weeks.
I always say no because A - my tests are always negative, and B - I don't want my appointment cancelled or rescheduled because they think I'm testing because of symptoms or exposure rather than condition of employment and I don't feel like explaining it to them.
Make it easier on yourself and tell them what they need to know not what they want.