I don't think there is. Batman is usually associated with more recent eras. The character's lore and themes fit better in modern or near-modern settings.
Batman becomes more complex. He has to adapt to new threats in Gotham, which makes him grow as a hero. His relationships, especially with his sidekicks, evolve. For example, his bond with Robin might face new challenges.
Some of his gadgets might not work as effectively in Themyscira. For example, his grappling hook might not find as many suitable places to attach to in the more natural and less urban landscape. But he could also adapt them. Maybe he uses his batarangs to cut through the thick vines or uses his utility belt items in creative ways to communicate with the Amazons.
She could be devastated. Wonder Woman might go into a period of mourning, isolating herself from others as she grapples with the loss. She was used to having Batman as an ally, and his death would leave a big hole in her life.
Batman would probably first try to contain the situation. He'd use his resources as Bruce Wayne to spread misinformation or counter the revelation. He might also go on the offensive against Scarecrow to stop him from spreading the news further.
Batman might become more confident in dealing with threats beyond Gotham. The power ring gives him abilities he didn't have before, and this could lead to a change in his self - assurance. He may also develop a more open - minded attitude towards the wider universe and the different kinds of powers and beings out there.
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.
Well, in many of these fanfictions, Batman's character is explored through his interactions with the League of Assassins. His strategic thinking is on full display as he tries to counter the League's plans. His physical and mental strength is also a big part of it. Facing the highly skilled assassins of the League, he has to use all of his resources. And his sense of justice is constantly being redefined in the face of the League's different view of the world. These fanfictions provide a unique platform to see different facets of Batman's character that might not be as fully explored in the official comics or movies.
Well, having Joker's blood could potentially drive Batman to the edge of his sanity. The Joker is known for his chaos and madness, and his blood in Batman's veins might make Batman question his own moral compass. He could start having darker thoughts or be more prone to impulsive actions.