Well, I kinda disagree with the "don't respond" camp. No, don't respond if you're going to be emotional or expect some validation back. But if you just want to throw something out there that may possibly defuse some of the misinformation, consider something calm, reasoned, like this:
"Dear SIL - Please remember that you are only hearing one side of the story. I've been working night and day to keep my kids in their home and food on the table. I did not choose this divorce, but I am left primarily responsible for the emotional and financial fallout and how it affects my boys.
Unfortunately, in a divorce, both sides suffer a loss of living standards. I am not asking for anything in the divorce that is not standard in the divorce laws of this country.
I too am concerned about H and what is happening with him, but he has made it clear that he does not want me in his life. I hope that you and the rest of the family will continue to look after him."
Stick to the high road, don't expect a response, or ignore a response if it's stupid. But putting this out there is not unreasonable, in my book, and if you're lucky, it might make her pause and realize she IS only hearing one side of the story.