I’m interested in pursuing an M.Mus degree but I’m in the process of completing a non-music bachelor’s degree. Under the assumption that my playing ability is up-to-par with candidates coming from music bachelor’s programs, how possible would it be to get into an average music program? How could I present myself to schools as someone who is seriously interested in pursuing the advanced degree?
Do you have a teacher at your college?
It’s really about the audition, not the undergrad degree. So if you have a great audition you should not have a problem.
I agree with glassharmonica. If you have kept up your playing with lessons, practice and performance, by all means go for it 
I personally know someone with a Bachelors in Business Admin who continued to study their instrument, gained admission to a very well known music school for MM performance, graduated, and works solely as a professional musician today. It can be done.