is not really part of the name of the school.
Nobody says they went to "Brigham Young University" when in fact they went to "Brigham Young University - Hawaii". I've never encountered that. They always mention the Hawaii part to distinguish it from its Provo counterpart.
Yet I've seen many, many times where the "I" is left off when students refer to what school they are attending. And if you have spent any time on campus up there people frequently leave the "I" or "Idaho" off when referring to the name of the school.