Here are a few things to consider:
1. Did you enter, or were you able to introduce evidence at court that the braces were not medically necessary? Judges have to make their decisions based on evidence, and an appeal of a judges decision will only take into account what is in the court record. If something isn't introduced as evidence, it isn't in the courts record.
2. For what it may cost you to fight this decision in hiring a lawyer/appealing, you may be able to take care of the medical obligation. If you do want to fight this, you should contact a local attorney ASAP especially if the judge has ruled and you seek a rehearing, reconsideration, or appeal.
3. If you are worried about this becoming an ongoing issue, you can ask for an injunction to prevent them from getting non-emergency medical/dental treatment that has not been deemed medically necessary until they come to court first (going forward into the future).
4. I have a link below from the Office of Child Support regarding Medical Support FAQs. With a court order they will collect an uninsured medical debt.
Every legal matter is fact specific, and there are often nuances in every case. This is intended for comment only, and does not create an attorney client relationship.