No. YANBU. The birthday girl's treat is the restaurant with her name, where she will feel like a mini celeb and they're sure to fuss over her. Dinner anywhere else would be pointless.
DM had her invitation and declined as was her right. Now she's changed her mind, too late, and want to upset her DGD's treat? No, no, no.
What would your DD learn if you did that - that Granny's whim is more important than her birthday? That parents break promises with impunity and children can do nothing about it?
I agree with the PP who says if your DS is so exercised about DM missing out, she can offer her place. If she won't, she has to pipe down. And whoever does come with you cannot sit with a face like thunder souring the atmosphere, either.
Since when is a child's birthday celebration about them, not the child?
You just need a loop saying, "This is not about you!"