Sorry this is long i really need advice

So im 37 weeks pregnant by my exboyfriend. I ended it because he was treating me like shit(name calling other emotional things) i broke up with him when i was about 8 weeks. He started dating again right away. (no big deal me and him were only together 4 months) but than he started stright out ingoring me and when we did talk he would call me dumb or tell how hes trying to get pregnant with his current girlfriend or blaming me, when i would give him updates about the baby. He messaged me that he didnt want up dates because he already knew he wouldn't know his son. I worked hard to keep him involved and try to settle the custody part before the birth with complete failure. I told him last month that i was done trying to talk to him and make it work. And we havent spoken sence. His whole family and him want him to have all his rights and everything but i dont. He wants to try and take my newborn for overnights.....witch i know wont work out... Hes in a band and often plays shows on weekends and drinks a lot. Im now choosing between not putting his name on the birth certificate he wouldn't have to pay child support and we wouldn't have to go to court uness he actually wanted to and i wouldn't have to risk him taking my new born away from me and i wouldn't have to deal with his poor treatment of me. Or i can put his name on it go to court and file the custody and child support and deal with all his bull shit..... What would you do? And why?