What do i need to do to get my childs father on the birth certificate if hes underage?

I'm 19 and he's 17, he's been there from day one. However he was not put on the birth certificate because he was underage and his mother wasn't at the hospital at the time we did her birth certificate to co-sign with him. They said she had to be there for him to be on the birth certificate. I would just like to know about what steps to take to get him on there. She has my last name due to us not being married. and as I'm 19 I have full custody of her, which is not a problem for me. He just wants to be on the birth certificate. My daughter lives with me and I take care of her and he gives money when he can and comes to see her almost everyday. He's never had her on his own because I've had the fear of leaving her and he wasn't ever there during the night due to other circumstances. I would like to eventually work on giving him that time but I don't think he's ready I think he would just let his mother do everything and I don't want that because I'm totally capable of doing what I need to do for my daughter. So advice would be greatly appreciated. We live in Memphis TN. Thanks for your time