I want my SO out

Let me start out by saying my relationship with this "man" is very complicated and though I'd like this to be short, it's not.
Long story short me and him do not get along. Once I found out I was pregnant The guy I knew for 5 years changed almost 360. He was a verbal dick and was manipulating me and emotionally and physically abusing me throughout my whole entire pregnancy until I cut him off at 8 months due to my preeclampsia and him sending my blood pressure into the 180's almost everyday. There was no way to report the abuse, I was unable and had no viable proof rather than a few text messages telling me to kill myself.  I do not spend any time with him, I do not live with him and do not contact him unless it's related to our daughter. We had made an agreement together that he would pay a certain amount of money a month, not through child support But through monthly payments at a reasonable amount.
For the first month things were going alright until this month all of the sudden there was no payment no phone call or text. A week went by and I called him and said what's going on? You have not paid yet? He then told me I could go fuck myself and that he will not pay a single penny anymore to support the child at all. He stated he doesn't care if the child was his or not and that he doesn't want to "waste his money." 
He knew that if he did not pay the set amount we discussed that I would file for child support. He abused me and I never reported him, and I refuse to combine any form of finances with him or "do it together." So the next day came and I asked him to meet me at my home to discuss the issue. I explained to him the next step because he outright refused the FAIR payment plan we had worked out together. He came to my home obviously high, smelled like alcohol and like he hasn't showered in days. His hair was matted and hadn't been combed and his clothes were stained and covered in yellow spots, along with reeking of weed. 
I explained to him child support was my next step because he cannot take any responsibility for the child. He then said If I want him on child support he wants custody. I wanted to punch him in the face. I would never let him even hold my newborn child. He drives under the influence, smokes weed and cigarettes in his home and does all other sorts of drugs and let's strangers into the home and leaves his door unlocked at night for people to come and go.
I don't want to go through a custody battle at all. But there's no way I can prevent him from getting custody of my daughter. I don't even want him near her. He's told me many times in saved text messages that he hopes I kill myself and he hopes I die while I was pregnant. He has also said he doesn't care about the baby and doesn't want to do anything for her. She would be neglected in his care, she could honestly die, because I used to live with him and it's no place for even an adult to live. 
Does anyone have any advice on what my next steps   should be? I live with my parents, I work a full time job at 9 months pregnant, I attend the university working on a bachelors degree, and I have supplied everything for my child up to this point he has bought nothing but a bottle from dollar tree, even though I told him I would be EBF. Do I continue with filing the child support? Should I just drop him? I am financially stable and would not be in any disarray if he left. Any advice would be helpful. Thank you.