Is going through a Chapter 5-13 with a MEB possible?

A chapter won't trigger a MEB, your BH therapist will need to make a recommendation to your PCM that you fall outside of AR 40-501 and that you should go through the MEB. MEBs are never automatic, a Dr. has to recommend you to the IDES process and after a review, they will accept you or say more treatment/therapy/whatever is required. You can easily be chaptered for the 5-13 because it's administrative and just needs higher authorities to sign off on it.

AR 635-200, Section 5-13 is where you can reference. But, if I understood it correctly, yea there may be an evaluation board convened.

The only chapter that can coincide with a MEB is misconduct chapters. The MEB and the chapter will go to the CG and he/she will decide what route to take. If your MEB comes back and says your fit for duty but do have a personality disorder, THEN the 5-13 chapter could happen.