Advice: Shut up.
What i mean by that is. Stop arguing with stupid. He's a clear class A twat, and you're arguing with him as though stupid is going to understand you. Stupid doesn't care what you think, because he's stupid.
Every time you react and get upset, stupid gets a lovely ego boost knowing that he's affecting you so much. His mind is "awww she's bothered by it, i can get to her, i can fuck up her day".
So shut up (meant nicely not horribly).
You tell him that if he wants to see his daughter, then sort it out and come and see her. I'd be dragging HIM through court for custody which I'd say is slightly more pressing than Michael McIntyre.
Yes he's taking his new cheap bit on the side out and about, how else is a guy gonna get laid?
One thing i've learned about guys like this though is that when they're like it with you, they'll definitely revert to type after the honeymoon period is over with anyone else. Bet on it.
Focus on yourself and your child, leave stupid to do as stupid does. Your child should be your priority, not arguing with your piece of shit ex.