You are going to be challenged about your vaxx status at some point so... Make the issue about the need to show papers rather than your vaxx status.
Tell whoever wants to see the proof that it is a private medical issue which you refuse to disclose. This applies to people vaxxed or not.
Many good citizens have been vaxxed, so what?
For example. let's say you have purchased season tickets to The Raiders in good faith. Show up to the game and when challenged for your vaxx "passport" refuse to provide on grounds of medical privacy.
If turned away, and after you have left - avoiding an altercation - contact them and demand a refund. Document everything and retain. You'll need it for the class action lawsuits that will be coming.