My DD was a little older than this when we let her dye her hair. If I remember correctly, she was in middle school.
Her hair has been pink, blue, and purple.
I have a funny story to tell. When her hair was pink, we had breakfast at Cinderella Castle. She had her picture taken with the prince during breakfast and he asked her what her name was.
Well, that evening, we were watching the parade in the Magic Kingdom. The prince rode by, looked right at our daughter, waved and yelled her name. Hi Sarah.
There's no way he would have ever remember her or picked her out in a crowd without the bright pink hair.
Fast forward to today. Her hair is back to it's regular color brown and she is a college student with fond memories of her special "prince moment and picture."
I don't see the problem with letting your daughter dye her hair. Believe me, there are much bigger battles to come in the future.