I'm starting to fear a never-ending, energy-sapping, GPA-sabotaging argument where agreeing to disagree is literally not an option.

There isn't a strong expectation here that people should never agree to disagree - see this old discussion, or this one.

That being said, persistent disagreement is a warning sign that at least one side isn't being perfectly rational (which covers both things like "too attached to one's self-image as a contrarian" and like "doesn't know how to spell out explicitly the reasons for his belief").