Make it a private matter between a citizen and a government. Better yet, don't call it a fine, call it a conscientious alternative. Like, a mandatory community service instead of a draft - a provision every humane conscription law should allow.
There is significant number of people for whom it is a very deep, primal issue, issue of integrity, of conscience, to the point of inability to leave with oneself after submitting to such request. Some napkin math: by UK stats there was ~300000 positive cases in the last 7 days; at the same time little over 5000 hospital admissions. So, the probability to occupy the proverbial bed is ~1.7%. The probability to catch covid within a year is not 100% either, and not even 50%, but let's assume it is 100%. Average cost of hospital stay - dunno, $100 grand? $200? $300? Let someone pay a 1.7% of this plus some on top once and call it settled. Make it a one-off tax surcharge. Heck, make it even annual surcharge, conditional on pandemic duration and vaccination status. But stop moralizing and dividing the society. People won't be happy with it, but they won't be as angry or suicidal, they won't feel desperate and cornered.