Nope, sorry, he laid hands on your child and screamed in his face. He can't be trusted around your children. All social niceties are off. Tell MIL you will see her at some other time around the holidays or she (and only she) can come to your house.
Speaking as someone whose family did avoid inlaw holiday gatherings for two years because of a problem relative, trust me when I say that making this sort of stand, and putting your own family/feelings first is liberating and totally worth it. Your BIL is the elephant in the room. You're all expected to tiptoe around the elephant and not mention the elephant's insane behavior, because that will make the elephant rampage. You can't reason with him. You can't expect rational behavior from an elephant. And the pattern does not change because an elephant never forgets how to control the people around him.
First, DH needs to stop trying to "work things out" with BIL. BIL wants DH to chase his approval and apologize and grovel because it puts BIL in a position of power. DH needs to take that power back. BIL doesn't have time in his schedule to meet? Fine. BIL doesn't want to talk about the real issues? Fine, allow him to blame his bad behavior on the actions of small children. That makes him look awesome. He doesn't want to apologize? Fine.
Second, you need to stop going to family gatherings where BIL will be present. Does it suck that you could temporarily lose contact with some family members while BIL gets to enjoy parties/events without consequence? Yes. But it does two things. 1) It shows family members the consequences of denying/enabling BIL's bad behavior (the consequence being that you are not present and they are stuck dealing with BIL alone) and 2) It takes away BIL's argument that all problems are based on your sons' bullying his. (Really? A 4 and 5 year old are bullying a 7 year old? Really?)
Third, if you do go to a family gathering, don't wait/expect/hope BIL will acknowledge you. This is what he wants, to hold some power of approval/validation over you. Ignore him as he ignores you. It's the worst thing you can do to someone like this.
Fourth, don't allow other family members to pressure you into "making nice" with BIL. This is the peril of being the "reasonable" sibling, you're expected to make all of the compromises. BIL had made it clear he doesn't want to see you or the boys anymore. You're just giving him what he wants.
You would not believe how much easier and calm life if once you draw the line with these personalities. MIL may be upset that all of her family isn't getting together at once, but you've got to put your kids and family first. Good luck.