That baby IS, so is due loving care. It doesn't matter what the biological mother is about if she is pursuing sin instead of responsibility as concerning that child. If that baby has a caring mothering woman, both mother and child are blessed. I take prayer to the issue of the biological mother first, asking that people be dispatched to minister to her for her salvation. Knowing her name to speak it out is unnecessary, as God already knows about her, and is looking for hedge-builders for her and that baby. He might be compelled to eventually say at her judgment He never knew her, but He will make the adoption family receive blessing in this life and the next.
My wife and I adopted a 5 day-old girl 34 years ago after "trying" to have a baby for 13 years. Her nurses in the hospital had already named her the name we chose while driving there to receive her. We had asked for a boy, had his name ready. Regardless of our desire, she was our first child, followed a year later by our biological daughter. We were blessed, have no complaint.
In all that I learned the amazing love of God who adopted me, a terrible Gentile bound for hell off and on. Had we not adopted that child there is no knowing of the consequences of some alternate set of choices with their consequences. I know I would have probably missed God and come away with a twisted version of His love. But now I understand.
God loves so much that He will adopt all the unwanted that choose Him, letting even a 5 day old baby choose as best she knows to. She grew up to seek her biological mother out, finding her, and determining to make sure that woman makes the right decisions.