I need thoughts and opinions please

My daughter is 5 weeks. Her father, he was my boyfriend, was not helping me with her at all. He would complain every night about her waking up. He was truly evil, even while I was giving birth. He never helped me when I came home postpartum for my own rest and recovery. It’s truly been a nightmare. We have a house together, and the baby was actually his idea. Well, he cheated on me in between having the baby and now. Apparently because he couldn’t wait the 6 weeks to have sex with me. I caught him and kicked him out immediately. This was a few days ago. Where he is at now, is a disgusting house with roaches and overcrowded with several kids sharing beds, etc. (His other children’s mothers house; who also has talked so badly about my baby). I decided I would have to get food stamps and whatever government assistance I could get. I realized I still did not have a birth certificate. So I looked through my hospital paperwork and saw that we never filled out the birth certificate. What a shock that was to see. Possibly a blessing in disguise. So now this is where I need help. I need to fill out the birth certificate form...

I cannot even think about him ever taking my child to that house or anywhere away from me. He is already bringing it up and actually told me that when he would pick her up, he would just give her formula since I’m nursing. Doesn’t work like that, buddy. 🙄

So I gave him this option. And I honestly feel like he doesn’t want to admit it, but he’s on edge thinking about taking this option but only if I continue to push it. —- him not signing birth certificate and having nothing to do with her. I wouldn’t go for child support or expect anything from him but he would never be allowed to have anything to do with her. Aka he would be signing over his rights.

Now what he is saying is that he would like to be in her life because “she’s his daughter so it is his responsibility”. To me that sounds like he feels like he has to. He said he would buy things she needs or send me money here and there. He said he would understand that I don’t want her leaving my side till her first birthday and that he would only come to visit or pick up and drop off from daycare. Could I use that help?? Yes, I’m sure I could. But what’s on my mind, is the day he comes to pick her up and take her to who knows where! I cannot even begin to think about that. Also, if he changes his mind and goes for 50-50 custody. I could not deal with that.

I’m getting mixed answers from friends so please I need anyone and everyone’s opinions.

Keep pushing him to sign off all rights & leave the father blank on the birth certificate

OR

Just leave him off birth certificate & see if he actually cares enough to visit her and help with her expenses

OR

Accept his help, be cordial with him, let him visit her & learn to co-parent with him

OR

Don’t trust what he’s saying, put him on child support or start the custody battle now