Good for you! I'm happy you see what's happening here and want to take a stand.
At least your ex isn't too smart, since she's giving away her playbook with her comments to you.
"I'm the one who gave birth to her": she wants to contest that you're D's parent or have any right to make decisions for her
"I felt scared": I'm thinking about accusing you of violence.
"I'm registering her at another school": She's doing this to try to make sure D is with her during the weeks, which would mean more overnights, which would mean more support from you.
Can you register D for the school previously agreed to? If W wants to play dirty, then you can get dirty.
If this isn't able to be undone, could you still do 50/50 with D registered at the other school, or would you have to move?