Cheers Shouty!
I think to be able to get to being friends, you actually DO need some distance/detachment.
I hope I explain this properly. I'll try anyway.
The only way of having a relationship in the future with the father of your DC is to END THIS ONE NOW.
He has lied, cheated and betrayed you all. He is not sorry, only sorry he got caught. He's biding his time only. he thinks that you are being a bit silly for now, and will let him back in your life when the little woman has cooled off a little.
He's not taking you seriously, both because you are sending him mixed messages, but also because he has zero respect for your feelings, doesn't think you are entitled to feel hurt at his treachery, and is dismissive of you and your family.
If you end it now, even get divorced, it'll send a clear and loud message that you and your feelings are a real force to be reckoned with, WILL be listened to or he will cease to be a part of your day to day life.
You will of course continue to co-parent, and for that to happen properly you need to be civil, nothing more. He won't need to know any details of your life, and you won't care about any of his.
IF you forge a new relationship, founded on love, fidelity, honesty, openness, trust, mutual respect and consideration, then there is a chance you can make it.
Without a break, without some healing time for you, it'll be hard. At the moment there is a total fog of grey, where you don't know what's going on, and it's in HIS best interests NOT to know what's going on either, cos then he doesn't have to admit/own up to losing his family for the sake of his dick.
As it's the school holidays, there are more options available for child care, perhaps your DC have school friends that they can go and play with and you can have them over when you are not working? or an holiday activity club through the school. Unless you give this guy a real taste of life without you and the benefits of family, how is it all going to be brought home to him the magnitude of his actions?
He needs to be banished from decent family life, and made to earn his way back in. If he really wants it, if he truly deserves it, he will do it, if not, nothing worth a damn is lost, is it?