I think this is the best possible solution, great write up and explanation. A minor improvement would be to make the card some kind of OATH device to generate TOTP tokens rather than a single ID number, so that you can reuse the same identifying token in multiple places with no way to connect the token.
Edit: On second thought, I can't think of a way to make that work, without compromising privacy, and I can think of a few possible ways that the original idea could potentially go wrong, too. Still, I think this is the closest possible solution.