I feel for you op, what an awful situation. And I can absolutely see why you are angry with her. I think the whole “but she wasn’t married to you” trope is utter rubbish. She knew he was married, she knew you and she knew he had small children. It takes…a particular type of character to get involved with a man in that situation and it does not reflect well on her. I also don’t think you are 100% blaming her. You aren’t stupid, you know this is on him and he’s a totally selfish asshole. Unfortunately, you also know he is their father, so as much as you might not want to see him ever again, you have no choice, because of your young dc. He is important to the children, she is not.
But, here is the thing. The OW is important to him. There is no justice in this, it is entirely unfair, but he can choose to be with whomever he wants. And the fact that he is so lacking in moral fibre and has picked a similar person to be in a relationship with is his problem. He threw his wife and children out of their home, shacked up with a barmaid then doesn’t bother to see them other than the bare minimum? He has to live with that, that is his choice, his lack of character, his shitty behaviour. It is a reflection of him as a person and says nothing about you. Children are not stupid and it will all come home to roost for him as they get older.
if you have genuine safeguarding concerns about how the children are treated in his care then take legal steps to protect them. If actually you are just mad as hell with both of them and are trying to “punish” him, don’t use the children for that. Instead, get some therapy to help you work through this massive betrayal.
I wish you well op and I think in time you will realise you are well rid of this rancid man.