I agree with the others, send her a
certified letter stating that you are more than willing to pay for braces but that the medical portion of the reason for the braces is the traditional kind. If she wants cosmetic ones than that is not a medical/dental reason to have them, and she would bear that cost.
Now the real issue is if she brings this to court. You can ask the dentist to give you a letter stating the dental/medical reason for the braces and the cost of traditional braces. Get the billing department to give you that amount. Is this dentist in the medical/dental plan? If you find one in the plan -even cheaper. Get that price. Then file a modification of
child support-medial to include "father requests that the mother stay in network for braces and that traditional braces be used instead of the cosmetic ones to reduce out of pocket for both parents. Father requests that if mother goes out of network or goes ahead with the cosmetic braces, that the father be responsible for 50% of traditional braces of an in-network dr which is in the amount of $ (according to Dr. X).
If there are other children, get this worded for all the kids and put in a line that braces will be paid one time for each child.