You should amend your court order to include who claims whom on their taxes. If they have a history of splitting, it is lame of her to claim both without saying anything, but it is a situation that can be corrected before next year. You'll lose out on last year and be a wash in lawyer fees for this year, but it won't happen again... It really is a shame when they can't just get along. 😕
Was the agreement in writing? I would look for a child support calculator for the state where you would file the court mandated child support and see what it would be. Not going through the courts could lead to her going to child support and claiming back child support and saying you never paid her anything. Then you have to prove it was for child support and not a gift. I would get it in a court order who claims that kids on taxes and ask tag you get to claim them both next year since you couldn’t claim one this year.
I guess you’d have to weigh if it would work out better for you to get courts involved. Would he have to pay more, less or about the same in child support? If he would have to pay more, would it be worth it to request to claim one child? If he would pay less or the same amount of child support, it would probably make sense to fight for him claiming one child.