Can I get some opinions on this??

My daughters “dad” and I broke up while I was pregnant because he was emotionally/mentally abusive and refused to grow up to get a job and provide. He didn’t talk to me my whole pregnancy and since the baby was born he has seen her twice. She is over a month old. He will go weeks without checking in or calling. Yesterday he told me he would like to babysit my daughter with his mom for a few hours. She is breastfed eats pretty much hourly so I said now is not a good time for that. She has never been away from me. I know the only reason he offered is because his mom wanted to see her, not him. Plus, he’s never made an effort to come over and see her so why would I send her there? I told him until she’s older he is welcome in my home to see her anytime. Anyways he got upset and said I never tell him anything about her and he always has to be the one to ask (which he should, I am taking care of a baby all alone) and then I told him how much I would love if we could have a relationship built on mutual respect for each other and have our main focus be the happiness of our daughter regardless of how we feel about each other. He is very immature and so he decided to tell me they I am a bad mother and that I need to be telling him about her day to day life and that he shouldn’t have to be the one to text me. I then told him that he should be the one reaching out, visiting often, and providing for her. He hasn’t provided a single thing for her, nor does he pay child support. I told him I will not stand in the way of him having a good relationship with her but I want him to show up consistently if he is going to have that relationship, not for one hour a month. He then said that he would come see her more but he can’t stand to be around me for more than an hour. The times he has been here I have been very kind and let him have his time with the baby so I’m not sure where they is even coming from. He has never taken care of a child, let alone a baby, and I would be terrified to leave her in his care alone because he is the type to do something stupid that could harm her. He is mean and tries to belittle me when he doesn’t get his way and I’m so over letting him make me feel like this. What are your opinions on this?

Also, I have multiple screenshots and call recordings of him being abusive towards me. I also have a log of how involved he was in the pregnancy(not at all) and a log of how often he checks in and visits. His family is also very abusive towards each other and I have some videos of his parents screaming and throwing things at each other. He still lives with them. As well of a list of things he has done that are scary and show how mentally ill and unstable he is. I have these things in case he tries to take me to court ( I don’t think he will, but I worry often). I have money to provide for her, a stable home, and a very healthy environment for her to grow up in. Has anyone had a similar experience? How did the judge rule?