I didn't have kids (thank god) with my cheater. I simply left his sorry ass.
We kept up a dysfunctional push and pull for a couple of months after I left, but his continued deceit and lying pretty much put the last nail in the coffin for me.
The one thing I absolutely refused to do was lower my expectations or blur my boundaries just to have him back in my life.
NO man is worth doing that.
Your best bet is to cut him off at the knees. If you continually let him know how much you love and miss him, then that just gives him a soft place to land and he knows he's always welcome to come home if things don't work out with his side dish.
So cut him OFF.
Since those are his kids too, don't let him get away with gratuitous 2 hour visits once or twice a week and then complete freedom to live like a single guy the REST of the time while you're caring for everyone the other 98% of the time.
If I were you, I'd come up with a fair child support amount per month that he needs to pay, and I'd also come up with a custody agreement that gives YOU a break as well as makes him act like a father and not some irresponsible party boy on the loose.
Lastly, ONLY talk to him about child related issues. No, "I miss you...." no, "I love you...." NONE OF THAT. Tell him that's ALL you'll discuss with him whether it's via text, speaking or email. Everything else is OFF the table.
I'm not kidding. Those guys who are allowed too much freedom to come and go into the family home as they please, and who are constantly being told how much they're loved and missed and wanted back whenever they come sniffing around are just being handed permission to continue doing exactly what they're doing - getting the emotional satisfaction of having their families while simultaneously living the life of a bachelor.
He can't have BOTH.
Cut him off at the knees.