Augusta State University, with one of its schools being the former Medical College of Georgia, makes perfect sense and should placate those associated with both schools.
I believe this is what the Medical College of Virginia did when it was brought into Virginia Commonwealth University.
Besides, people call a school what they want, and a long descriptive name just doesn’t hack it. Example: Who knows Virginia Tech is really Virginia Polytechnic Institute and State University?
Someone else can come up with the names of other schools under ASU, but just to feed the fire, I suggest the Arsenal College of Arts and the Bartram College of Science from the illustrious list of five.