Although I don't know if I would try to bribe (I'm risk-averse to stuff like that), I have heard of healthcare workers who are, shall we say, "flexible".
Work through all possible options, and if it comes right down to it, and you need to eventually make that appointment to get the jab, I would show up at the medical office, be really nice to people, roll up your sleeve, and then say, "Wait a minute. I really don't want this. Can we ... you know ... just skip the actual jab? All I really need is paperwork."
There really is a chance that you'll get lucky. And why not try? You're at the office to get a jab -- a last ditch effort to avoid it is worth it and you have nothing to lose.
Your scenario is worth a try, but if the healthcare person is reluctant, a discreet offer of cash might make the difference.