Am I in the wrong?

This will be a long post so I apologise in advance

I’m having trouble with my ex at the moment. Backstory: we split a few months ago, during our relationship he was abusive and would gaslight me. He was selfish and everything had to be his way or there would be an argument. Walking on egg shells trying not to piss him off that kinda thing. Anyway, he kicked me and the kids out and got with someone else almost immediately. He usually has her on weekends although sometimes lately he doesn’t. He also usually visits every Wednesday but the trouble started last week when I realised he lied about not having to see her. He didn’t show up all week, didn’t message or anything to see how she was. To some it may seem like nothing but it’s unusual for us because he always did before. He ended up saying he had to work nights so that’s why he didn’t turn up (I don’t know why he couldn’t just let me know so didn’t wait in). That was the last of it, then I saw my friend who casually mentioned in conversation that she saw him on the Wednesday in the car with his new girlfriend. If it was any other day it would mean nothing to me because it’s his life and nothing to do with me. But she saw them together at the same time and day that he was supposed to be seeing his daughter but couldn’t because he ‘had work’. Now the problem for me isn’t the fact that he was with his new girlfriend because obviously they’re gonna spend time together, the fact that he lied about why he didn’t turn up to see his child is what I have a problem with, or the fact that he couldn’t spare any time for his daughter but could find time for her. Children always come first IMO. So even after I found this out I decided I wasn’t even going to say anything it’s wasn’t worth the drama. Until today! He randomly turned up at my house after not seeing his daughter or having any contact for almost 2 weeks (he also didn’t turn up for her on the weekend) l, he was here for about 30 minutes and in that time he sat down on his phone for most of the 30 mins. My daughter is 2 years old and going through the tantrum stage, she started having one of her tantrums and he stood up and said “I’m going now, she’s getting on my nerves”! That’s when I saw red! I though it was a shitty thing to say just because he can’t cope with his child throwing a tantrum (all toddlers go through it). So I said “I have her 24/7 and have learned to deal with it and calm her down, you’ve been here for half hour and and that’s all she’s getting from you?” He then goes on to call me a dickhead and I need to shut the fuck up, and called me a little fuckwit. Same old name calling as always, he won’t change that. So then I decided to bring up why he lied about working so he could be with his girlfriend instead of his child. He laughed in my face and said he wasn’t there (the place where he was seen), then he said he was there but his girlfriend wasn’t with him (she was) and then he just called me more names, laughed in my face and left. So I messaged him and asked him to just not lie just to get out of seeing his child, and please put her first because she should be his priority like she is mine. I told him the name calling has to stop and all I wanted was for him to be consistent with seeing her. He messaged back to say it’s hard for him not waking up to her every morning and putting her to bed at night but I can’t help that! He’s the only person who’s responsible for that because he threw us out onto the street because he found someone else! I told him it was all his decision and this was what he wanted, you make your bed you lay init. I also told him I never told him he had to come and visit her at my house, I haven’t got that right. He’s her parent just as I am and he was more than welcome to come pick her up and take her somewhere every Wednesday on his visits. It just never happened like that and kinda fell into the routine of him coming over. All the split and moving out was all his idea, I had no choice but to go along with everything and now that I’m living in my own house happily without him it’s suddenly my fault why he can’t tuck his daughter in at night or wake up to her every morning? He says all this but can’t even handle her for 30 mins and leaves when she screams! In the end I just told him to turn up on his weekends to pick her up and that’s it, no more house visits because IMO we cannot co parent amicably. Am I wrong? There’s more to this story but I’ve gone on long enough

* no we haven’t gone through court. We’ve always wanted to deal with things on our own. Even the child support arrangement. The amount he’s supposed to give me (we worked it out ourselves online) was quite a lot because he’s earns good money but to make his life easier I suggested he pay half! Ive been more than fair to him

*jennifer court will be the next step but I can’t afford it at the moment, and he knows it! But I will be pushing for the higher child support. He will hate me for it because he’s very money oriented but it’s for his daughter not me