R’ Moshe writes that a woman being a Mashgiach for a kashrus organization is a problem of serarah, unless she only acts as the eyes of the Mashgiach and does not enforce the rules herself (and even then, it’s possible that that would only be good enough in the bedieved case R’ Moshe was dealing with). It’s hard for me to see how being a Rabbi or Rebbi should be any better.
