Your right with most of it. Still she could be more understanding with husband. He is probably worried for his sister. Plus who would want to tell a very ill family member that her dog would be rehomed?

Yes SIL is an AH for not training the dog properly. But can you blame husband that he rather not discuss with his sister just now while she is still in recovery?

That only shows he is a decent person who wants to support family when its needed. And for this OP could indeed be a bit more understanding and empathic even if dog needs to be rehomed.

I'm not saying she must support husband fully especially if she is afraid of the dog. But to be so harsh to husband is simply AHish in such a situation. She could have handled this way better.