Not exactly. Batman Begins is one interpretation of Batman's origin story, but it's not the definitive or only 'real' one. Different Batman stories and adaptations offer various takes.
One idea could be that the OC is a childhood friend of Bruce's who has a unique skill, like being an expert in ancient martial arts from a far - off land. They could influence Bruce's training and view on justice in a different way. For example, they might introduce him to more spiritual aspects of combat and self - discovery.
Yes, Batman Begins can be considered a mystery fiction to some extent. It has elements of mystery as Bruce Wayne uncovers secrets and faces hidden threats in Gotham City.
One idea could be exploring Bruce Wayne's early training in more detail. Show his struggles and how he overcame them. For example, his first encounters with Ra's al Ghul's League of Shadows could be more in - depth, with new challenges and characters added.
Perhaps a plot twist where Bruce Wayne discovers that his father had a secret life as a crime - fighter too, but in a very different way. It could change Bruce's perspective on his own mission and add a family - related mystery to the fanfiction. This could involve finding hidden journals or messages from his father that lead him in new directions.
One idea could be exploring Batman's initial reaction to being part of the Justice League in a 'Batman Begins' context. Maybe he's hesitant at first, still haunted by his past traumas from 'Batman Begins' and having to learn to trust these super - powered beings. He could be constantly assessing the League's motives and methods, which could lead to some internal conflicts within the team.