Sorry, am going to be cruel to be kind here: your mum sounds quite manipulative, controlling and bad at understanding and respecting boundaries. Telling them home truths (ie being bitchy, petty and cruel, because that's what 'home truths' means), telling them she will be visiting and when, instead of asking or waiting to be invited, arguing but then still expecting them to come to the dinner party... it sounds like she's used to throwing her weight around and being able to say and do whatever she wants and you and the rest of your family are used to going along with it and keeping her sweet. I'm sure she can be lovely otherwise, but she sounds an awful MIL and a very demanding, controlling mother who doesn't seem to entirely view her child(ren) as full adults.
The very fact that you are on here trying to work out how to make everything nice again and make your mum happy again, shows that you are in a pattern of automatically feeling the need to make your mother happy, be responsible for her moods and feelings, etc. The fact that you think that them not going to a dinner party after being insulted is as bad a 'crime' as how your Dm has treated them speaks volumes.
Your brother was right not to go to the dinner party after arguing, IMHO. This shows a mature sense of personal boundaries and self-respect. If he has been insulted and upset then why should he stuff it all inside himself and pretend it never happened and go play nicey-nicey at mummy's house just so she can save face? Them not going actually sounds like a hard lesson learnt for her (or it should be) - she can't expect to say and do whatever she likes and have people ask 'how high?' when she says jump.
Your focus should be on your brother, not SIL. Your mother is his mother, not hers. He needs to sort it out - if he wants to. For all you know, your brother might find your mother incredibly hard work and controlling, etc., and looks to your SIL to support him in keeping contact brief and infrequent. She might be suggesting he make more contact with his mum and he's refusing, for all you know.
If your mum really wants things to be sorted out - as opposed to having them apologise and her feel like a vindicated victim who has been in the right all the time - then she needs to extend the olive branch, examine her own behaviour in all this, apologise with genuine contrition and understanding of what she did wrong and then look at how to sort out her patterns of behaviour that make her upset them so. Yes, your brother could do the same too, but from what you say, even though you're on her side, she sounds like she behaves quite selfishly and thoughtlessly a lot of the time. If someone treats you that way, it's healthy to want to reduce or cut contact with them, even if they are your own motheer.