Do I let my daughter dad come to the hospital. Long post

My little girls father left me shortly after becoming pregnant he never wanted kids and was adamant that I get an abortion. Fast forward he decided he want to be apart of her life went to one Doctor appointment the 20 week and sent a check for baby stuff that didnt even amount to a quarter of the cost of what has been gotten for her but I was ok with that at least he did somthing. I had thought even thow I'm angry and hurt for him leaving I will be civil for my daughter and have kept him in the loop let him know everything that is going on . It's been ok but at times rocky like when it comes to her name ( I believe this is my decision he chose to bow out and be a sometimes maybe dad ) or that hes only been to one appointment because of distance and finances but yet he took two weeks and went to California ( again hes not helped a whole lot financially either but once ). Fast forward I'm 33 weeks now still arguing over her name, im way more emotional and scared that I wont know what to do when she gets here and terrified that I'll fail as a single mom with no help after she comes home so more angry that he bowed out. But during a conversation about appointment and name when I told him he has no say in her name because hes the one who left but at least I've heard his opinion he got nasty said some mean things and told me I have not seen mean yet and that he has a say. In the same conversation as he us demanding what he has the right to he tells me that after she is born he wants a paternity test. He wants a say but believes she may not be his are you kidding me and want to be there for the birth. I'm angry and hurt and sad. First the threat of me not seeing him mean yet then that. I'm sorry but its insulting. I was so in love with that man and see such a different controlling and mean side of him since the split.

So my question would you let him be at the hospital after questioning paternity or five him any decisions making or information until he gets his stupid paternity tes. I did tell him I will do a prenatal DNA test since it's not invasive just a blood sample he said no it's to expensive that way.