why can't they admit they're approximations to something else, rather than come up with this totally new, counter-intuitive epistemology where it's not allowed to assign probabilities to fixed but unknown parameters
Because they don't accept the premises of Cox's theorem -- in particular, the one that says that the plausibility of a claim shall be represented by a single real number. I'm thinking of Deborah Mayo here (referenced upthread).
Have you tried offering de Finetti's choice to them? I had a go at one probability-resister here and basically they squirmed like a fish on a hook.
http://xkcd.com/1132/
Is this a fair representation of frequentists versus bayesians? I feel like every time the topic comes up, 'Bayesian statistics' is an applause light for me, and I'm not sure why I'm supposed to be applauding.