How should I go about publishing a manuscript in which my mentor does not actively support my best interests?
First, regardless of what you do, secure your independence of her in every way: funding, reference letters, etc.  All that has to run smoothly without her.

Second, forget her.  Don't count on her finishing. She won't do that at all or if she does it, it won't happen personally, but she hires another student.

Third, prepare a journal submission, and get someone senior to read your paper and provide critical comments.  You need a more-or-less constant eye on the paper from some senior person in the area to guide you away from pitfalls you don't see due to the lack of experience.  If you don't have that person, count on a rejection of your first submission.

Fourth, put your prior, no longer supportive advisor either into acknowledgements or as the last coauthor, depending on how much of her contribution is left by the time point of submission. Submit the paper.