If you were part of a Family Court proceeding, you would essentially need to subpoena the dental records and possibly hire an expert witness to prove that they were not necessary. I am not sure where the Group A/B argument fits in.
I believe that you may have more of an argument that your spouse did the procedure without your input, but this would largely depend upon the wording of your agreement.
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If a professional states that it is necessary for braces, then in all likelihood the court will order that the braces are necessary.
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Let me see if I understand this fact pattern. You are a non-custodial parent with input authority. This means your custodial opponent has to seek your input and if she fails to do so, she is on the hook. Further, you are asking whether something other than basic healthcare/dental care imposes 50/50 liability. That's up to the court likely based upon testimony of an expert witness who can render an opinion as to whether a particular procedure was "necessary".
I have been through many of these kinds of cases. I would recommend paying the bill. The natural outcome is that the child will demand other things and the opponent will no doubt talk down about you and your denial of payments for doctors' procedures. This will hurt your relationship with the child and cause future outbursts and unpleasant conversations.