Hermione's character could develop a sense of independence. Away from her usual friends at Hogwarts, she has to rely on herself more. She might become more confident in her own abilities without always having Harry and Ron by her side. Also, she could become more diplomatic as she has to interact with students from different backgrounds at Beauxbatons. This could help her in future situations where she needs to negotiate or work with different groups in the wizarding world.
In this fanfiction, Hermione's character might develop a new side to her personality. Since Beauxbatons has a different social structure, she may become more of a leader. At Hogwarts, she was often overshadowed by Harry's fame. But at Beauxbatons, she could be recognized for her intelligence and leadership skills right away. This newfound recognition could boost her self - esteem and make her more assertive. She might also start to question some of the things she took for granted at Hogwarts and develop her own unique magical theories based on her experiences at Beauxbatons.
Her character would likely become more adaptable. At Beauxbatons, she'd be exposed to new ideas and ways of doing magic. This would make her more open - minded. For instance, she might learn different ways of casting spells that are more elegant in the Beauxbatons style, which could change her approach to magic.