I hope most are agreed that "hyperbole" is not a good answer for the original question.
I'm thinking the sentiment expressed in the question is close to a definition of "dogmatic," though "dogmatic" does not connote anything about being extreme, more about simply stating opinions as if they were facts. The question pertains as much to the arguing from a particular point of view as it does to the expression of that point of view. "Dogmatic" seems to have this overtone, since the employment of dogma is almost exclusively in the context of persuasion, and not infrequently, argumentation. Anyway, it follows that "to argue dogmatically" would be very close to the second part of "arguing emotively from an extreme, or biased, point of view as if your view is fact," but it does not capture the "extreme" aspect the poser wants, more the "arguing as if your view were already ascertained fact" aspect.
So, "dogmatic" is not a perfect fit--but maybe there isn't one.