L.M.
I agree with you (and I have been the bio-mom carrying coverage even when ex was ordered to because he often quit jobs and therefore lost coverage and even when he had coverage we kept secondary insurance)...the coverage is for the child's health needs and having the dual coverage helps both parents by reducing total out of pocket expenses (so while they have the coverage to help them, it helps both sets of parents).
I suggest that dad go to the doctor's office, with court order to show what they both responsible for. Be sure doctor knows that there is dual coverage and that after both insurances have paid, he will be responsible for half of the balance. They would know what the out of pocket expense would be (or could find out) and you would get the most objective/unbiased info.