Of course he does. He does not want to get strapped helping you pay for the child he created for the next two decades. He wants you to bear that burden.
Unfortunately for him, there is no paper where he can just "sign away his rights" because its convenient for him to do so. He is stuck. Smart move on your part getting custody and you are getting child support too.
Whether he exercises his visitations or custody rights, he is financially obligated to support his child. He can have zero visits and can refuse to see the child but he is obligated to support the child. Visitation and child support are not swapped for each other and are not related.
Contact DCSE
Division of Child Support Enforcement 1-800-468-8894
Email at AskDCSE