I work at a dental college, so let me see if I can help.........
I know it's obvious to you and me that it appears braces are medically necessary, but you know the drill.......get it in writing from the professional. If it is proved that the braces are medically necessary....and you have an order that all costs are split, then you have an argument for whether the NCP has to pay.
But since BM is the NCP and appears unwilling to do this for her child, then I strongly recommend that all the appts. be made while the child is with you. Other than that, if she still refuses to seek tx for the child, then your only recourse is court. As for the cost (and she still refuses to pay it), the court will have to be notified about that as well. Not sure how it's done in your state, but some states have all med. reimbursements done thru the CS office, so that may be where you have to start on that issue.
Your other enemy here is time......with kids' teeth, a few months can make a big difference because changes can happen pretty fast. And the last thing you want is to have to resort to oral surgery in order to alleviate the situation if it goes from bad to worse.