It sounds like you also go running to him to tell your side, though.
I think you need to really reduce how much you're talking to him. Grey rock him. Stop feeding into the drama.
He gets emails from the school, you know this. So you don''t need to tell him school stuff. Move to a co-parenting app or message only, and just do factual stuff, don't get into discussions. Stop playing his game.
If she goes running to daddy, so what? Let her.
The good news about splitting up with the guy is that you don't have to listen to him any more. It's literally the best bit of a break up, start enjoying it. 😁
At your house, it's your rules. At your house, she picks up her clothes. However you choose to discipline, is between you and her, and you just broken record her about that. At dad's house, it's his rules.
Just say to her honestly that you're tired of being played off against her dad, and this is how it is. Stop playing her game.