I think you're going to have to suck it up and take this one.
Despite what the conspiracy people say, this isn't some big plot by the government. It was a shitty situation that under any normal circumstances you wouldn't have needed to be told to get a vaccine when it became available and you would have just gone and gotten it.
You can still chose not get vaccinated. That's still a freedom you have, but with any choice there are outcomes and unfortunately for you in this case, the outcome is find another job.
I don't know whose proof you need but your natural immunity, what that may be, will likely wear off and probably isn't as great as you think. You survived, good on you, but hundreds of thousands didn't. The FDA cleared the vaccine, so other than your ego, I'd say if your job is important to you, get the shot and fight the political battle another time.