I've an English (and theatre) BA and am working on my MFA. However, I
have no real skill at speaking/reading Japanese yet.
How much of a working knowledge of Japanese would one be expected to
have?
I would assume a conversational fluency at least

Liam,
You don't need any Japanese to teach English.  Japanese is prohibited in the
classroom in most cases.
You need it for daily survival.  How much depends on where you live.

And what are the chances that the English I taught could be at a
University level, or corporate?
With your degree, zero chance for university.  That type of job requires at
least a master's degree in a related field like linguistics, plus refereed
publications.
For corporate work, your chances are 50-50.  Depends on the recruiter.  You
will likely need one to get your foot in the door.