I am not sure in the state of La. However in Alabama, they would not allow me to put my son's father's name on the BC, b/c I was tech. married to my now EH. I of course did not want my EH's name on there. We had been seperated for more than a year. So on my son's BC, it reads under Father's info: MOTHER REFUSED
When we went to court for my divorce, I had to have a DNA test proving that my EH was not the father and even though my son's father was not on the BC, he still had to pay child support and had visitation rights.
But I did have a DNA test to prove he was the father. Which cost us $500 out of pocket.
Of course we are married now and the custody papers are void.
Hope this helps