Hats off to you for how well you have responded to the posts on this thread. Whatever happens it’s clear you have very good mental health.
I’m wondering if your sil cannot cope with you being stronger than her. Up till now you’ve always been in a place of weakness in relation to her. Ie She was the haver of child and you were the have not. Your dh did warn you she can be very tricky. Now you’re seeing this for yourself.
Your mil did your sil no favours. She was the golden child and wore the crown of producer of first grandchild. Now her life is down the pan and her disliked brother has everything she wants. A shiny new baby and a solid marriage to a fab woman. Not communicating with you imo is most likely linked to her feelings toward herself - and view of who she should be as well as her feelings toward her brother and you.
Your mil chose to divide and conquer her children, to create petty jealousy and deceit. It isn’t surprising that your sil is reverting to type when under extreme pressure.
It is also hard to be a golden child. Im not trying to paint her a total victim for an adult your sil is choosing her behaviour. However if her parenting was anything like mine (scapegoat here) I had so many gaps in my knowledge. So she may not know how to act differently.
If you want to write the card to her as suggested by Shasta I would go for something very simple. Imagine how you’d communicate to a child for if she has gaps in her knowledge of how to adult, she is in some ways still a child. Something like “I am sorry you are hurting. I love you. I miss you. I want to be there for you through this difficult time in your life. I want you to be there to share my joy. I want our children to love eachother as much as I love you.” Then leave her to digest. A written card is very different to a text btw.
And no sending of scan. I see your reasoning. That will create the opposite effect. Ultimately you may have to accept this isn’t fixable. The situation is about her, not you.
One last thing, do keep an eye on how your child is treated in the family. Your dh is not the favoured child. My mother and brother tried to make my dd scapegoat by association. It didn’t work.