I know it's easy to snap in the heat if the moment and shout, but that doesn't sound like what's happening here, shouting I. The heat of the moment through frustration and exhaustion isn't done as active discipline method, it's an impulse action IMO pm with your dp, it sounds like it's a regular thing and I think it's easy to forget how intimidating it can be to a child to have a grown adult, esp a grown adult male shouting at you regularly. What is it that dd is doing that requires him to step in and discipline her in this way when you don't? Why isn't he following the lead from you?
Does he have DC of his own? Does he "discipline" them in the same way and are they also scared of him? His discipline clearly isn't working if he's shouting at her often enough for her to be scared of him. So what's his next method to try? If he was the boss at work, would he discipline colleagues the same way. If not why not?
Dh has raised his voice at his son twice in the 17 years I've been with him, (both for safety reasons rather than discipline) his son wasn't scared as he had built up a relationship with him since birth and knew his Dad was safe while testing boundaries, had his Mums partner raised his voice at him, or even myself, I can see him being scared as he wouldnt know their boundaries as they a new adult in their life, they haven't had the years of safety testing boundaries like they have with their own parent. Hope I'm making sense.
My own Dad would always be shouting at my Mum and us, they split when I was about three but I remember her often standing in front of me as my dad was in my face shouting at me for not putting a doll away, to which he'd start shouting at her for being to soft. I quickly learnt that Dad would use shouting as an intimidation method, it's so hard to describe but the shouting in its own isn't what made be scared of him, it was his interactions with my mum, the change in atmosphere when he came home, seeing my Mum change from carefree to nervous and on edge when he walked in. The bending over backwards to not make him shout. I hated the few times he'd be drunk as it would be worse and he's sometimes push my Mum if she tried to stop him shouting at me. I really can't articulate but despite my Dad never laying a finger on me, I was very much more scared of him than the future boyfriend who hit me. It's only after his death a few years ago that I'm seeing just much damage he has done and how scared of him I was right into adulthood.
Shouting doesn't always have to be abusive, but your Daughter feels scared of him and she couldn't come to you to talk about it and spoke to school instead. Does he act intimidating in it her areas, does he expect things a certain way and everyone has to follow his rules type thing? If he's not abusive he will be horrified that he's made a little girl scared of him and want to change that and work on it. If he's anything like my own father he will be angry that she's spoke of her feelings, that she's making a fuss, that she's lying to split you up, that you and her should be greatful that he's being a father figure (cam paring himself to your ex if he's not around) that you dont discipline her properly so he has to.
His reaction to learning he is scaring her will say a lot.