FWIW, I've been co-parenting with Ex for 6yrs now (3 of them prior to courts involvement). I'm an adult and don't have birthday parties any more. Though, my family does do a little Sunday dinner for me on my birthday weekend. But they've always planned it on a weekend where I was going to have my D. I always make my plans based around the parenting schedule. Before ex moved away, she would actually ask if I could watch D on her bday weekend so she could part the night away. To each their own.
Your situation....To keep the peace, I'd allow her to have the time. Maybe you guys getting back to a civil relationship can start today. Just be sure to get full awareness that you will be taking make up time on your bday or the following weekend. Or better yet, just ask if she'd like to switch the entire weekend.
OH - Residential parent of D5 after a long distance move by Ex. If your child(ren) aren't a priority, you will have no chance at gaining custody.