We definitely do have that power to refuse, and we patients do need to remember that, and feel completely free to exercise it, if and whenever, we want to. Even if you are at the appointment to have it, when you find out something they do with it, that you do not want.
I have refused, more that a few times over the past years.
One of those that I decided to refuse once right there, I did decide to reschedule later on, after I had more fully researched the methods and reasons and options. I was then more comfortable with deciding to have it.
Another one, I decided to have done later, at a different Provider that would listen to my requests regarding it.
The others, I decided not to have at all, and was very glad not to have them. Other doctors agreed, I didn't need them!
On the other hand, and in the OP's post, I think, we sometimes decide or feel it is the best thing for our own health, after we have tried other methods, to have the unwanted and likely un-needed test or procedure, for the reason that: the doctors and/or insurance, etc, might refuse to give us the needed procedures that we do want, until after we've had the ones that they want first.
A bad spot to be in, but then, we have to decide between our options.