Here's why and how it's presented in the story:
* It's stated in the manga and anime: The series never suggests Hatsuharu dyes his hair or that it's anything other than his natural hair. It's simply an unusual trait.
* Family traits can vary: The Sohma family is subject to strange curses and transformations. It's not outside the realm of possibility that genetics could manifest in different ways, including unusual hair color.
* Hatsuharu's Dual Personality: Hatsuharu has a "Black" and "White" personality. His hair reflecting these dual traits is symbolic and accepted within the context of the story.
* Acceptance within the group: Other characters in the story acknowledge his hair color, but they don't express disbelief or require convincing. It's just part of who he is.
In conclusion, the premise of Hatsuharu needing to convince anyone about his hair color being natural is incorrect. It's presented as an inherent part of his character and is accepted as such within the narrative of Fruits Basket.