Your ex still has rights to his biological child, regardless of whether or not he is on the birth certificate. You should discuss visitation issues with him. No, he is not likely to gain “custody” but he is likely to get visitation. His lack of contact with the child will hurt him when it comes to custody.
If you don’t have a lot of money, you should call your local DSS office and tell them you need to establish child support. He is likely to then ask for a DNA test through the courts, which it appears you want anyway. Once the DNA test proves he is the father, you need to file an action for child custody and child support.
There are self serve centers in many counties which are set up to help people pursue these types of actions on their own. You may also want to contact Legal Aid for assistance getting started, but they have limited resources right now to handle any family law issues. They often will have pro se clinics on child custody, so you should call your local office and see if they are having a clinic anytime soon. You should sign up if they are.