Yanbu to be fuming. But I would approach this in a different way.
It seems that your ex very much repeats everything that your DD has said about his gf back to her, and that is never going to end well. FWIW my eXH did similar. My DS didn’t like his GF, but he refused to hear it from me, and insisted that DS tell him himself, so he did, and then ex told his GF word for word what DS had said.
Like your ex, XH and his GF argued, had screaming matches, she had a DD who they insisted was DS stepsister from the outset and essentially insisted that they behave as siblings, and then she fell pregnant with their own baby early on.
I told DS that I was always here to talk to. That I would only ever support his having a relationship with his dad and would always encourage him to go, as when you lose a relationship that early on it can be hard to rebuild if you want to in the future. He was also 10 when this all started.
By the time he was 13 I believed he was old enough to make his own decisions, and more and more he started staying with me during the week rather than going there, until they went on holiday and again she and DS had a screaming row,she had a go at DS because he wasn’t acting in a way she thought he should, then she hurled a lot of insults about me across a crowded restaurant, it was so mortifying that even her own daughter left.
After that DS never stayed there again.
He’s 19 now and has a vague relationship with his dad. He talks to him etc but they only see each other maybe every 3 months. He has 0 relationship with the GF or the GF’s DD, and 0 relationship really with his half sibling. He sees him maybe twice a year.
And all of this could have been avoided if eXH had been a bit less cavalier about it all in the beginning, and had actually put his DS’ feelings first.
So what I would say to your ex is that while you understand he has a GF who he wants to spend time with, every time there is an incident like this, he is doing harm to his relationship with his child, and that one day, when she is old enough (and that will be one day soon,) she will be able to decide not to see him, and he will have no control over that and neither will you. So you suggest that perhaps they meet at her GM’s house in future, so that the relationship between them still has a chance, while she’s still young, and before it’s too late.