Need advice

So I never really put my age or my boyfriends age into thought when making decisions regarding my labor and delivery. We are both teens. Well my aunt who was a teen Mom herself is saying I shouldn't let my boyfriend sign the birth certificate or have the band to get the baby out of the nursery. I feel like that would be wrong for me to do so because he is the biological father and it could start unnecessary drama. I might be able to stop him from signing it at the hospital but he could easily just go get a DNA test and get it put on. And with the band it's his child. I feel like its a right for him to have that. Shes saying if he signs the birth certificate that hell try to fight for custody if we break up. Me and him have both talked about what if we break up and what would happen with our daughter and we've agreed on a co parenting method just in case. Neither of us want to go to court but ik we would if necessary but I feel like it will never be necessary bc we are pretty civilized people and have already came to a mutual agreement. If we did go to court id get sole custody because A.our state favors the mother B.i have better income C. He has disabilities and legally saw as he can't take care of himself. So I'm really not even worried about court even if it did happen. Would it be best if I let him sign it and get the band? I really don't 100% understand what my aunt issue is bc it's not like I can really stop him from being on the birth certificate or anything I feel like it'll just be drama if I tried to stop him and I don't see the point in why I'd need to stop him in the first place.