I've been the SIL in such a situation. Maybe my story will help shine a light in another direction.
The offense that led to me drawing boundaries with my SIL (H's sister) was the last of many straws. She felt threatened by us coming to live nearby, which we did because FIL was very sick. And it was a cross continental move, done purely to be there for them all as FIL ailed. SIL lived in the village near her parents (we were in a neighbouring country 100 km away), but she felt that we were going to usurp her favoured status with her parents (lots of favouritism by PILs, they were actually not nice people) and as soon as we araived, she started icing me out passively-aggressively and - we later found - gossiping about us negatively to all and sundry, especially PIL.
When she finally iced me out so obviously that I asked her privately what was going on, she twisted the story to everyone - the WHOLE family - to make it seem like I'd attacked her verbally and been mean to her. She was crying and saying how hard she had tried to make me feel welcome, and here I was attacking her. Cue my bewildered explanations to my ILs, who had already been primed by her to think I was a troublemaker who had it out for SIL (I realise now, it was projection). I was, after all, a foreigner, so it was easy for them to think I was "crazy" and believe all kinds of nonsense about me.
My H tried to straddle the fence for a while and kept visiting with the kids (with my best wishes) but I stopped going to visit, since I had zero energy for this conflict - I was still trying to recover from the huge and utterly exhausting move, establishing my career in a new continent, learning the language, helping the kids settle, making friends and contacts, figuring out financial rules etc. I hoped my SIL would settle down and we could try a rapprochement in a year or two, but things just got worse and worse, and ended with SIL so blatantly exerting her PA icing tactics on our eldest daughter that my H could no longer overlook what she was doing, and then HE called her out.
I of course got ALL the blame, PILs were convinced I was poisoning his mind against them (egged on by SIL).
I too was deemed "coercive", because me not continuing to visit them - and being upset at the unkind way I had been treated by my SIL - meant that I was trying to force my H to stop visiting.
I was also called narcissistic, and it was noted that I had had problems with my father (because he was a serial cheater who treated my mother heinously), so that made me a mentally unstable rabblerouser and family destroyer.
The PILs got so aggressive and angry about me that my H didn't want the kids around them and stopped taking them.
So then the story was that I was trying to weaponise the children against them.
PILs became so unpleasant that H too eventually stopped seeing them.
Of course, that was my fault too. I was an absolute demon to them, horns and tail and pitchfork. I had cruelly destroyed the family, taken the beloved son away, taken the grandchildren away, I was a complete madwoman who needed to be locked up.
So yeah, when I hear stories of, "my SIL is narcissistic, has blown a simple thing completely out of proportion, she's coercive, she's crazy" - I'm always a little doubtful. Because I've been in your SIL's shoes and the stories spun about me had NO bearing on reality.
Maybe it really is as minor a matter as you say. But you seem argumentative and very determined to be right. Why did you send that message to your brother, about how wife was trying to get between you and him? That's meddling and trying to get between him and her! This is the kind of thing MY SIL did too. And it didn't make me feel safe with her at all.
You have come here with your version of the story and got the answers that you want, which is that you're SIL's victim and that yes, SIL is a complete crazy bitch.
Have you considered that maybe this was not the first time you've upset SIL? That maybe you're not right?
Maybe you should give yourself time and don't react. Let the dust settle. Be honest with yourself.
And think about the family dynamics - is your family dysfunctional, is everyone truly accepting of and pleasant to the DILs? In my case, my H's brother's wife was also excluded and gossiped about by SIL whenever SIL wanted to get her knickers in a knot, but at least they were from the same country, so the treatment wasn't as bad as it was for me.