MILDrama "DH is also devastated. He's totally with me when it comes to her behaviour as he had a miserable childhood but has tried hard as it's his only remaining parent. We're trying to take a reasonable stand as she's been such a bully but it seems we're at an impasse."
OP, please stop being reasonable, if a boy or girl were bullying your child in the playground would you want to be reasonable, to faciliattae more bullying?
No.
Ask your dh to consider some counselling to recover from the shit of his childhood. Then decide to go low or no contact with her or simply to allow her to cut herself out of your lives.
"I'm not sure I understand semanwen's comment" That's because it is ridiculous.
"Plus we didn't use my father's name for my only son as we thought it would upset her!"
Please stop centering this woman in your lives, she does not deserve it.
MILDrama "My DH worries that this will tear the family apart, something he promised his dad he wouldn't let her (MIL) do." He is really not bound by a promise he made to his father which he may be powerless to do anything about. His duty now is to you and your children. Please get him to get some counselling to get free of this. His father may have requested this in good faith by your dh needs to be relieved of this duty, it is NOT his responsibility.
"My MIL has also said to me that she'll never forgive me for not letting her take my daughter home from the hospital when she was born?! How am I supposed to make that better??" You are not, you too must see she is either unhinged or very manipulative and really hot someone to be left around your kids unsupervised. I'd only want her in my life if she could behave.
Really, you and your dh need to centure your own kids, not wider family. The kids are what is important for you, they are children, the adults in the wider family may need to work this shit out by themselves.
You sound lovely, and kind, please save your energy for your own kids.