Advice in custody/child support

My husband had a child with a woman out of wedlock. She is legally still married and the child has her and her husbands last name. We have not gone after custody due to the Unwillingness of the biological mother to cooperate and do what's best for the child. We feel it would be in child's best interest for us to leave it alone for the time being as she will hold him over my husbands head and use him to get her way. She already speaks badly of her other children's father (her husband) to the children and make them feel guilty for wanting to spend time with him so it's best if we leave the situation alone for now. Just wondering if anyone else is going through a situation where u or ur significant other is not fighting for custody. Also could she come after my husband for support in 5 years even if he has not been deemed the biological or legal father or this child?