Yes, deny hospitalization to unvaccinated people, at least in publicly funded hospitals. This is a much fairer way to deal with it than a mandate that forces people to vaccinate under duress.
And stop worrying about the unvaccinated transmitting to the vaccinated. The risk to the vaccinated is sufficiently low that it should not be a dominating concern any way. Also, vaccination doesn't stop COVID transmission enough to create herd immunity even at near 100% coverage, so we have to resign to COVID being endemic, especially given how rapidly a single strand RNA virus mutates.
The caveat is we could potentially eliminate COVID, influenza and a host of other viral infections, if we take the reigns off the pharmaceutical industry, by eliminating FDA approval requirements for vaccines. Machine learning has made enormous paradigm shifting breakthroughs in other fields, like natural language processing and protein folding, and could very well do the same for drug discovery and testing. Drug development could potentially keep up with the viral mutation rate if it could subsist solely on faster processes, and jettison the slower FDA mandated ones.