The FAA tells controllers via Order JO 7110.65 "Once the alert is issued, it is solely the pilot's prerogative to determine what course of action, if any, will be taken." What the FAA is telling you per the AIM is incorrect.
Let's see, we have three FAA sources on this, the controller's manual, the regulation, and the AIM. Two of them agree, and one of them doesn't, and you're assuming that the one that doesn't agree with the others is the correct one. Seems kind of arbitrary.
If I flew an airplane like that in the face of conflicting instrument indications, I'd be dead by now.