I don't think there's much you can do unless and until you witness it yourself. You have the babysitter's take on things, but that's all. Maybe the sitter overreacted, maybe she was defensive if she perhaps wasn't paying close enough attention or failed to intervene if a situation was brewing. You have your kids' view of things, but you don't really know what your kids did (if anything) to precipitate a reaction. So stay open to other explanations.
The mom didn't sweep it under the rug - she imposed consequences. But she didn't see it either and of course the intervention/consequences were delayed, so they're not as impactful as something done at the time. (Plenty of kids are crafty enough to not do this stuff in front of a parent - which also means she can control her behavior a bit.
I think you handle it as you would anything else. If your kids don't want to play there, then they take a break, either total or reduced numbers of get togethers.. Kids don't all have to be buddies just because the parents are friends. Maybe all of the kids are sick of getting together so often, so it's fine to branch out.
Now, if you have a chance for a day trip together, then you'll be there (as will the other parents) if an incident occurs. Or maybe your presence will deter the behavior. If the kid has a problem that's developing, then at least seeing it will give you something to go on. You can still be sympathetic to your friend at having to go through a tough time with her kid. This will be good preparation for all of you for the teen years too!
Meantime, you work on building resilient children who can either shrug stuff off or learn some good coping strategies for dealing with difficult people and standing up for themselves. It's all a learning process and it's not perfect.
I have mixed feelings about a letter of apology (or an automatic "I'm sorry"). Yes, it's good to teach manners and all of that, but if the kid isn't really sorry, what good is an apology? Too many kids, especially young ones, think that an apology neutralizes the offense and makes it go away, like it never happened. An apology is not a do-over, the way many kids (and, frankly, many parents) think it is.
If you think this behavior is escalating, then you handle it as you would if it were any kid - you'd tell your child he/she doesn't have to play with that child. Make another plan. You don't have to get into it with the other parents - you just say your kid can't come today because she is doing something else. It's okay when kids change their friendships up a bit, and it's okay to give a little space to let things calm down. My kid has moved in different circles at different stages of his life, and it's fine if friendships drift away. The thing that happens if you confront your friend is that she's likely to take it as a critique of her parenting. And that never works out well.