Therefore, make sure you have rights to record from whichever publisher owns your arrangement of your fight song.
Just having permission from your campus is meaningless unless the arrangement rights are held by your campus. Figure out where your fight song came from, contact the rights holder, and get permission.
Finally, this isn't your job. You're not the cheer sponsor. They're the one who picked out the music, they're the one who should be doing the metaphorical legwork here.