The thing is, you might also rather he get hit by a bus when he goes out of the door this morning but that isn't going to happen either. Just try and remember that he's their dad, that's their story. You can't change that. Get some counselling for the controlling behaviour you've experienced, so you'll be less angry with him and inclined to seek justice behind the scenes.
Unless you have a decent relationship, I'd forget the goal of amicably co parenting, and settle for a parallel parenting situation instead. Get a court approved app, such as Two Houses, and use that for communication to keep the verbals to a minimum.
You need a child arrangement, and if you can't agree, you can go to court to get one, bearing in mind that it might not be what you want, so if you can arrange one out of court that you can bear, then that might be the way.
You don't get to "allow" what he does on his agreed time, or who he introduces the DC to, in the same way as he can't prescribe what happens on your time. If you feel there is a safeguarding issue that's different.
Keep your lip zipped about him, his GF, stay steady, loving and stable and truly prioritise them by keeping your feelings about their dad to one side. They are half him, and they know it, every time you slag him off you're slagging off that part of them too.