I have to say, your MIl sounds like a rather unpleasant loony. I agree with PP what she seems to be treating your BIL’s baby as the golden child. This is a common tactic among narcissists, and can be horribly damaging for the children involved. Have a read up on it if you haven’t already. Her invented stories that show her in a positive light as the devoted GM also suggest a narcissistic personality, so I would bear that in mind.
My father is a narcissist, and several months ago I went NC with him, as he had decided my DD was the golden child, while my poor DS was systematically ignored and excluded. This (among other reasons) led to my decision to cut him out of our lives in order to protect my DC. Obviously, the two children in your situation are cousins rather than siblings, so you may be able to employ damage limitation without going NC, but if I were you, I would keep my eye on things to make sure your MIL’s behaviour doesn’t have a negative impact on your DC.
I doubt there’s any point to you saying anything to the woman, to be honest, although you’d be perfectly within your rights to do so. If she is a narcissist, she will enjoy the drama of the confrontation, and it will likely encourage her.
With regards to your DH, I too would be cross at his refusal to acknowledge that what his DM is doing is very wrong. I doubt there’s anything he could do or say to change her behaviour, but it would be a huge comfort to you if he at least acknowledged that what she is doing is so clearly unacceptable. I think you must have a very calm and honest chat with him about your feelings, and ask him about his too, because this sort of resentment can build and could have disastrous consequences for you as a couple. But bear in mind, if this woman employed similar tactics while he was growing up, he will see all this as fairly normal, and it may be difficult to get him to see that it’s not. Best of luck, OP.
Oh, and ignore the PP who seems to think your MiL’s behaviour is perfectly acceptable. Your DC stands to be very badly hurt by this woman’s behaviour, so why should you just accept it?