4 month shared rental - possible to register as domicile?

Hey y'all! I did read the mod notes on this but still wasn't sure, so I figured I'd ask. I am interning in Germany next year with the Erasmus program, but my work is only from the end of March to end of July. With that in mind, will I be able to sign a housing contract for only 4 months? I think in shared apartments/houses it could be possible, right? In that instance, would I immediately just submit my move-out paperwork at the start of April? And can I register myself through that contract?

I did contact studentenwerk (since my internship is at a university research institute) and they said I can be put on the list but it isn't certain AND I would have to pay for 6 months, not 4 months. I get a small living stipend and it would be rather unpleasant to spend the money on months I won't be there.

Any advice/experience would be grand. I appreciate y'all!!