The child support office may not be able to get a judgement for the medical expenses for the kids, but you can by filing a contempt of court motion compelling him to pay, I would ask that a percentage gets garnished until it is paid off. As long as your ex lives in WY, WY has jurisdiction and that is where you will have to file. Check the guidelines for child support, if it says health insurance is an add-on, get his portion of what you pay for the kids added onto the child support and as that it is garnished from his wages. He is a governmental worker, he should be easy to find.

This is going to cost money, but since he is failing to comply with a court order, you may get the judge to order him to pay your legal fees. There is no guarantee of that, though. So you need to weigh if this fight would be worth it to you.