I watched a movie on HBO the other night, Contagion. It's nine years old but very prescient.
At the end of the film after a vaccine becomes available, people wear wristbands with barcodes that can be quickly scanned to show proof of vaccination.
I suspect something like these are in our futures.
The easiest way to gum up plans for private Covid passes is to allow paper documents... they are hoping for digital passes. For all entities that want proof, make them accept and store physical copies for several years.
Make it as much work as submitting insurance claims, so much so that dedicated personal will be needed to cross check, validate, and maintain an archive.
Some people would opt for digital processing but for those forced to be vaccinated, presenting paper documents would be a silent protest to intentionally screw with their bureaucratic system.