Get a fake card. They are easy to make. They do not deserve your loyalty. Remember, your card is a prop. Also act like “Oh of course I got the shot”. Shake your head at the idea someone wouldn’t.
It’ll buy you another year, but that might be enough.
If the digital proof comes up, deal with that then.
That's it, just weasel your way out of this.
Just don't take a moral stand against this absurd requirement; it's the 21st century, after all.