He could be a sort of vigilante in the wizarding world. The wizarding world has its own laws and authorities, but Batman might see that there are areas where they are lacking. So he would take it upon himself to patrol the wizarding areas, especially those that are not well - guarded by the Ministry of Magic. He would use his gadgets, which might be modified to work with magic, to take on dark wizards and other threats.
Batman might become a mentor to the young wizards. He has a strong sense of justice and discipline. He could train the students at Hogwarts in self - defense and how to think like a strategist. He would teach them how to use their magic in more practical and effective ways in combat situations, just as he trained Robin in his own world.
He could be a detective - like figure in the wizarding world. Since he's so good at solving mysteries in Gotham, he could use those skills to uncover dark secrets in Hogwarts or among the wizarding community. For example, he might figure out who is behind a series of mysterious disappearances in the Forbidden Forest.
He could be a protector. Just as he protects Gotham, he would protect Hogwarts and the wizarding world from dark forces. His skills in combat and detective work would be very useful. For instance, he could detect the presence of Death Eaters or other dark wizards much like he tracks down criminals in Gotham.
One possibility could be through some sort of magical accident. Maybe a powerful dark magic spell gone wrong in the Harry Potter world that somehow reaches across dimensions and pulls Batman's soul into the HP universe. Another way could be through the will of a powerful magical being. For example, a wizard or witch with god - like powers decides to reincarnate Batman for their own purposes, perhaps to bring a different kind of justice to the wizarding world.
Batman's brooding and serious personality would likely make Harry more cautious. Batman is all about planning and preparation, and Harry would pick up on that. For example, Harry might start to analyze magical situations more thoroughly before acting, instead of just relying on his instincts.
It could be really exciting. Maybe Harry would use his magic in Gotham City to fight crime alongside Batman. Or perhaps they'd discover their relation through some mysterious magical - technological means. Their different skills and abilities would make for an interesting combination in the story.
He could be a powerful antagonist. Since Anubis is associated with death in Egyptian mythology, he might be interested in the power of the Deathly Hallows. He could try to steal them from Harry, believing they would give him more control over death in the wizarding world.
Fem Harry Potter might be a strategist. Given her experiences in fighting Voldemort and his followers, she has a wealth of knowledge about facing powerful and cunning enemies. She could come up with plans that combine both magic and the Avengers' combat skills to take on threats.
It could be really interesting. Harry might grow up with Batman's sense of justice and combat skills. He'd probably be more stealthy and less reliant on magic in some situations. Maybe he'd use gadgets like Batman instead of wands sometimes.
If Harry Potter meets Batman in fanfiction, it would be an interesting crossover. Harry has his wand and his knowledge of magic, which is so different from Batman's high - tech gadgets and martial arts skills. Batman might be initially skeptical about magic, but Harry could prove its power to him. They could end up on a mission together, perhaps to rescue hostages taken by a villain who is using both magic and technology. Harry could use magic to get past magical barriers, while Batman uses his skills to fight off any non - magical threats. It would be a story full of action, adventure, and a bit of cultural exchange between two very different fictional universes.
Batman could be a mediator between the Greek gods and mortals. Since he is both a human with a strong sense of morality and a powerful figure, he could stand up for the rights of mortals against the often - tyrannical rule of the gods. He might negotiate treaties or agreements to ensure that the gods don't overstep their bounds when it comes to interfering in human affairs.
Logistically, his Hydra form would be large and difficult to hide. In Hogwarts, where secrecy is important, it would be tough for him to keep his Animagus form a secret. Also, if he uses his Hydra form in battles, he has to be careful not to cause too much destruction as the Hydra's natural instincts might be more aggressive compared to his human form.