The first Rapunzel stories were Persian and the girl in it had raven black hair. Just because the Grimms brothers made their version blonde it doesn't mean her hair has to be blonde. They only wrote down one version based on what they wanted to tell but there were many other versions told by mouth. I can tell you with certainty that any girl with super long hair gets nicknamed Rapunzel even if she has other hair colors.
So your brunette rapunzel is just as authentic as any blonde one. She just needs a tower and very long hair.