How does she express the feeling of hate towards him? Can you share a couple of examples of situations that bring on the stress?
Her behavior may seem as antagonizing and that may very well be her intent. Another possibility is that through her acting out, she is testing him (to see if he would reject discipline her)
Regardless of her trigger (because she's doing this purposefully whatever her reason is), he should not remain passive. He may try to redirect her attention onto something else such as :when she says hello fatty, he may say something like "I wonder what that label is supposed to mean, or he can joke Oh, I suppose I can work out and ask her if she works out, etc. He is not ignoring the behavior but at the same time is not feeding into it. Avoidance can imply or be interpreted by her as passive aggressive response.
The two of them may benefit from some bonding activities ex: if you would ever be able to entrust them with a task at the house or have them go to the story to buy something, etc. He can try to find at least 2 things to compliment her on and things that he encourages in her behavior ex: if she is good in a specific are in school for him to show interest in, if she is athletic, creative, etc. For the time being, continue to talk to her that he is not trying to be her dad, that she is not supposed to act like he is one (she can do that with her own dad) but simply to treat him the same way he treats her. When she does, even the slightest improvement can be praised by you (to reshape her behavior with positive reinforcement) If she is the youngest, she may be still developing her own coping ways (and that takes time) and ,that is why she reacts differently to him than the other kids.