I know this is just a side issue, but I notice that your ex is just your ex while the OW is a whore.
Your ex is the one who betrayed you, it was him who swore to love you and to foresake all others.
Call him the names he deserves! Might make you feel better if you unleash the rage where it truly deserves to be unleashed. HE promised to be faithful to you, she didn't. HE lied to your face, she didn't. HE lay in your bed, looked you in the eye, while knowing he was betraying you, she didn't.
I don't mean call him to his face, that won't be helpful! I don't mean tell your children, that won't be helpful. I just mean safely get it all out in some way. He is the Prince of Bastards. HE is the whore. Maybe you need to do that in order to be able to move on.
But as far as the children are concerned, hard as it is (and no matter how impressed I am by how reasonable and grown up everyone else is, I hold my hand up that I'd be locking myself in my room at night and indulging in some serious voodoo and shit if my husband crapped on me from a such great height, so I understand your anger. He made a fool of you. He betrayed you. It is very hard) you have got to seperate 'bastard husband whore' from 'father'. and do nothing that could harm your children.
Mind you, it sounds like he doesn't need any help in that department if he is refusing to see his children unless she can come along.
they're his children. He should want to see them, regardless. She shouldn't be a consideration.
You have got to find a way to let go of your anger if you ever want to be happy. He betrayed you. He's not worth you destroying yourself with bitterness.