Yes, there are quite a few. You can start by looking on fanfiction websites like Archive of Our Own. There are stories that explore Catwoman's relationship with Batman in the Arkham Knight universe in various ways, such as her adventures during the events of the game, or her backstory and how it ties into the overall Arkham Knight narrative.
Archive of Our Own (AO3) is a great place to start. It has a large collection of fanfiction, and you can easily search for Batman Arkham Knight Catwoman fanfiction there. Just use the search bar and type in relevant keywords.
In this fanfiction, it could be interesting if Catwoman discovers a secret about the Arkham Knight that Batman has overlooked. She could use this knowledge to her advantage, either to make a deal with the Arkham Knight or to help Batman. For example, she might find out that the Arkham Knight has a weakness related to a certain location in Gotham that was important to his past. Catwoman could then lead Batman there to take down the Arkham Knight.
For Batman, his character could be developed by showing more of his internal struggle. In the Arkham Knight scenario, he's facing a lot of challenges. Maybe he starts to question his no - kill rule more when he sees the destruction the Arkham Knight is causing. With Catwoman, her thieving skills could be explored in more detail. She could use her skills not just for personal gain but also to help the people of Gotham in a more subtle way.
The Arkham City DLC story is an extension of the already rich Arkham City universe. It might start with a seemingly minor event that gradually unfolds into a major crisis. There could be new gadgets or power - ups for Batman to use in the DLC, which are related to the story. For instance, if there's a new type of toxin spreading in Arkham City, Batman may need to find a way to counter it using a special gadget he discovers during the DLC's storyline. Also, the DLC might explore the relationships between different characters in a more in - depth way, like the connection between Batman and a lesser - known villain who has a big role in the DLC.
The Catwoman story in Arkham City is mainly about her navigating the dangerous and chaotic Arkham City. She has her own set of goals, often involving stealing valuable items. She also gets involved in the larger conflicts in Arkham City, interacting with other characters like Batman. For example, she might be after a particular jewel or artifact while also having to deal with the various gangs and super - villains that populate the city.
The Catwoman story in Arkham City isn't too lengthy. It adds an exciting dimension to the game, but you can finish it within a reasonable amount of time, depending on your gameplay style and how much you explore.
In the Arkham City DLC story, Batman is the central figure, as always. However, there may be other key characters as well. For example, Catwoman could be a major character in some DLCs, either as an ally or with her own motives that intersect with Batman's. Additionally, some lesser - known villains or anti - heroes might be introduced and become important in the DLC's storyline. Maybe a character like Hush, who has his own vendetta against Batman, could be a main character in the Arkham City DLC story, adding new layers of mystery and conflict.