"How do we know that a refusal of the offer of prayer isn't going to piss off a Christian who holds our lives in their hands?"

I think that IF I were an atheist and found myself in this situation that I would refuse as nicely as possible and then ask point blank if my refusal was going to make a difference in the quality of care I was about to receive. Put the offender on the spot. If I thought there was going to be a problem after that I would take it up with the proper authority/manager/superior.

But I might add that in my opinion a Christian that would change their quality of care based on either answer should not be considered any Christian at all. It's a complete reversal of the teachings that they profess to follow.