I just have to say I can name two churches that have refused or placed limits on the baptism of children. If you think it proper for me to name them I willl. The first was a Catholic church in PA. The mother was 17, the father 19, both raised Catholic, mother practicing, father not. Not married, no intentions of getting married. Both still living with their parents. When they went to the pastor, they were told that they would christen the baby, but the parents had to consider marriage. The pastor felt the child would not be raised in a sinful environment if they remained unmarried. The grandfather, (the mother’s dad) made a sizable contribution to the church and the baptism went off without a hitch!
The second was in NJ. The parents, married by a Lutheran minister. The mother born raised and practicing Catholic, the father a born and raised non-practicing Episcopal. Since the mother was the parent with a practicing religion they decided the children would be raised Catholic. They went to the Catholic church closest to their home. The pastor told them that they could not christen the child, because the parents were not married in the eyes of the church. However, the pastor would perform a “ceremonial blessing” for the usual donation of $250 and then they would be married in the eyes of the church and the baby could be baptized. The mother converted and now is an active and practicing Episcopal. The child was baptized Episcopal.
No matter what anyone says. The churches position varies from parish to parish.
By the way, I am a practicing involved Catholic. I am a lector at my church.