The normal Schengen study procedure is that the student takes the university documents to the police and is issued a temporary residence card, then when this expires the student gets an additional 90 days tourist time added on after. The expired residency card can be presented at exit to prove no overstay. They should be able to offer and assist with some kind of process like this, and it is the way to go. Trying to get some kind of extension later likely won't work out.