The Riddler could also be a major antagonist. He might be intrigued by the daughter and set up a series of elaborate riddles and traps for her. He wants to test her intelligence and see if she is worthy of being the offspring of Batman and Catwoman. His riddles could lead her all over Gotham, from the seedy underbelly to the tallest skyscrapers.
The Penguin could be a main antagonist. He might see the daughter as a new threat to his criminal operations in Gotham. Since she has the combined skills of Batman and Catwoman, she could easily disrupt his illegal business, like his smuggling operations.
The Joker would be a classic choice. His chaotic and unpredictable nature would pose a great threat to Batman's daughter. He might try to corrupt her or use her as a pawn in his evil schemes against Batman.
Maybe the daughter discovers a secret conspiracy in Gotham that involves both her parents' past enemies. She has to use her unique combination of skills, like Batman's detective work and Catwoman's acrobatic abilities, to uncover the truth. Along the way, she forms her own team of allies, some from the side of the law and some from the fringes.
In some fanfiction, their daughter could be a powerful force for change in Gotham. She might use her parents' influence and resources in unique ways. For example, she could start a new movement to clean up the city, using her mother's charm and her father's detective skills to expose corruption and bring about real change. This would be a fresh take on the Batman - Catwoman story and add a new layer of depth to the Gotham universe.
Ra's al Ghul could also be a main villain. He might see the child of Batman and Catwoman as a potential weapon or a threat to his plans for Gotham. He could try to either recruit the child to his side or eliminate them, which would put Batman and Catwoman in a difficult position as they fight to protect their family from Ra's al Ghul's devious schemes.
She could be a vigilante like her father, but with a more flamboyant style inspired by her mother. Imagine her swinging through Gotham in a purple and black catsuit, using her wits and combat skills to take down criminals.
The main antagonists could be those who envy her wisdom. They might be jealous of the attention she gets or the power that her wisdom gives her. So, they try to undermine her at every turn, spreading false rumors about her or trying to make her look foolish in front of important people.
It could be the old enemies of Jack Sparrow. They might see his daughter as a way to get revenge on him. For example, those he has double - crossed in the past.
The descendants of the Death Eaters would make great antagonists. They might still hold a grudge against Sirius Black for his actions against Voldemort and his followers. They could target his daughter to seek revenge or to prevent her from uncovering any secrets that Sirius might have known about them.
It could be a new group of rogue ninjas who oppose the ideals of the Hidden Leaf Village. They might target Kakashi's daughter because of her connection to Kakashi and Naruto, seeing her as a symbol of the village's new generation of strength.
The Joker could be a main antagonist. He would be fascinated by Harry's magic and would try to use it for his own evil purposes or just to cause chaos. He might try to drive a wedge between Harry and Batman by using mind games.