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Your husband has every right to as for shared custody, or at least visitation, even if he hasn't been in her life up until now. He needs to bring it up with the mediator at the mediation hearing. The mediator can include it in the report to the court, or in the court order if the mediator is doing the order. If the mediator doesn't want to include the issue of custody since it wasn't part of the mother's initial petition, then your husband just has to file a Motion for Shared Custody, using the same caption as the child support case. He can then ask for a hearing or mediation on the issue of shared custody.
He may still have to pay child support, depending upon how much money each parent makes and how much custody he gets. Keep in mind,m it is doubtful that your husband would get 50/50 custody right away since he hasn't established a relationship with the child but he should ask for it anyway. He can always go back and ask for it in the future after he has been the child's life for a while.
As for the mother not wanting him to see the child, he has a legal right to visitation if he doesn't get shared custody. She can't take that away from him unless there is something terrible involved such as violent behavior, drug abuse, etc. He would have to do something really bad to lose his parental rights to visitation.