I am automatically suspicious and it sounds hinkey, and I am sure that anyone who is in social services feels this way in general, of when ex husband and ex wife ( or father and mother of child) both adamantly want NO CHILD SUPPORT and no court involvement. Because sometimes, if she is allowed a choice in the matter, the wife will say she doesn't want child support, not because of her high minded principles about not wanting the government involved, or because she still wildly agrees with and has faith in the baby's other parent, but because the baby's other parent is a lunatic and she's afraid of him and he has bullied and threatened her into the idea that she just wants to get away and she's not going to dare to ask him for anything. There used to be a form in my state that the mother could sign saying that asking for child support from the father would be dangerous for her safety when applying for aid. Later, they fixed that nationwide. The state has the responsibility to collect from the other parent, not the person applying for the aid.
It's kind of like the old pressing charges in domestic violence cases, where it was taken out of the abused person's hands because you literally had the abuser standing there in cuffs screaming, "If you press charges against me, I swear I'll come back and make you sorry!" as they were taking them away. It doesn't sound like you were supporting your child, at least to the extent of providing insurance, or that you were about to do so until caught up in the Medicaid situation and are being collected upon.
I scoff at the idea that you had to wait to put the child on your insurance when she lost her job where she was obviously carrying the child on her insurance before this. Most insurances allow changes due to family emergencies. And it was to the point that the child ended up signed up on Medicaid. That means that you had an uninsured child floating around out there for a few days. That's how the program got involved. If I were the state, I'd be leery of letting you go your merry way saying WE want this whole case dropped, WE don't believe in child support, etc. when it very clearly sounds like WE is you. And you are a blusterer and a bully and an argu-er. If you go in blowing like this to DHS to try to get this dealt with your way, you may find yourself removed from the office by the police. You will have to pay for the coverage provided during the time your child was on Medicaid. Period.