Unfortunately, the LBH is limited when she is not living in the same house. I mean, once she is out from under his roof, she can pretty much do whatever she wants. If she's in an affair, she is dishonoring and disrespecting him, but they are already separated, so what else is there to do but maybe divorce.
I am sympathetic to parents who do not want their children around the affair partner. As a grandparent, I have had the sorrow of facing that situation, and it was hard. Unless the law says she cannot sleep with him while the minors are in the house, I don't know what the H could do about it. He might check to see what it would take to get a R.O. Someone else who has had experience in that area may be more helpful.
The sad truth is that once the couple separates, boundaries seldom come into the picture. The H can hold her feet to the fire about child care/visitation, and things of that sort. B/c she will take advantage if she thinks she can get away with it. He mainly needs to have something legal stating the days, etc., otherwise, it's just a source of argument.
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!
