Always follow the instructions of the particular school you are applying to. If that school (UNCSA) says they require a song from musical theatre, you better have a song from musical theatre.
I don't know much about choosing a song for an MT audition, but I suspect that performing the rap section from a song (such as Wonderland) isn't going to be what they are looking for. I agree with supportive that it won't show your singing range. It's almost like you have found a loophole and abused it, they want a musical theatre song, but you have found something that is (technically) part of a song which (technically) comes from a musical, but which sounds exactly like any other rap song instead of like a musical theatre song.
Also remember that even if they ask you to sing, it is still an acting audition. In musical theatre, the performers are still acting, still playing a character, when they sing. When choosing an audition song, and preparing the song, remember that you will still be evaluated on how you portray the character. (I'm sure this is why they are requiring an MT song).
If you are uncomfortable singing, find something that's easy to sing. Also remember that they are probably only expect you to sing 16 bars.
If you are just auditioning for acting, they may be more interested in having you sing so that they can see how you project your voice, how you breathe and so on than in your actual singing ability.