3 things in response

1. Again, the hospitals don't get to decide about the mandate. That's a federal policy enacted by CMS. Regardless of the creative solution you propose for these hospitals, they either comply or lose roughly 2/3 of their funding.

2. Please provide an example of a single vaccine mandate that allows for someone to opt out of because they can demonstrate they were previously infected. I've never heard of a school or university or hospital allowing people to opt out of a specific vaccine because they already had that disease.

I had pertussis in high school, but it doesn't mean I would get to opt out of a required DTAP booster if I were working for a hospital which required it.

3. Out of the many arguments I've heard against vaccine mandates is that it violates HIPAA for an employer to know basic medical information about their employees. Putting aside how that's a misreading or misunderstanding of HIPAA (it only applies to medical providers), wouldn't your proposed solution allowing for people to prove they have natural immunity require that the government have some big database of who has been sick with COVID?