While the composition may indeed be in the public domain, you also have to make sure the recording is as well. Beethoven has been dead for centuries but many of the symphonies have copyrights on their recordings.
The first thing you need to do is inquire with the school as to whether such a performance license ALREADY exists. When I was working with college broadcasting, the school had a blanket license that covered all manner of things from music-on-hold, to performances of the student musicians, to music at dances and the campus pub, to the campus radio station (originally carrier current like your closed system but later a public FM station... I had the distinction of filing the LARGEST power increase application ever done by a radio station when we went from 10 to 13,720 watts).