I think your SIL was being the reasonable one really.
She was worried, justifiably as it turns out, that her child was not safe in your mother's home.
Your mother had already told her to relax while she was supervising him, and then sent her away to the kitchen.
If SIL was so cautious beforehand, I can't see that she would leave him without an assurance from your mother that she would be watching him.
And then he pulled a bookcase over on top of himself and was hurt.
I would have been furious and shocked and upset if that was my DS. I know someone, now an adult, who was injured in a similar way as a child. He was left with brain damage, walks with a limp and has lost the use of one arm because of the bump to his head when the shelves fell on him.
To have my child put in that situation, where he could have been left brain damaged or killed, and then be dismissed as a vindictive bitch, would probably kill of my relationship with your mother and your sister.
It sounds like they already dislike your SIL, and already consider her to be a vindictive bitch, and are using her upset about this to prove to themselves that they are right.
You don't know if your SIL has decided to stop your mother seeing her grandchild.
She will still be feeling shocked and upset, and relieved as well that he wasn't more seriously injured. And relief can be just as extreme a feeling as shock and upset, and people express it in odd ways.
What should happen right now is that your mother accepts your SIL was right, her home is dangerous, she was dismissive of SIL's fears and feelings, and as a result her grandchild came to harm and it was her fault.
She needs to be apologising to your brother and SIL, not bitching with your sister about what a vindictive bitch her DIL is for objecting to her child being injured.
Your mother and sister are being unreasonable. Your mother is the one to blame.