Another good pairing could be with Lila. Lila is a master manipulator herself, and when paired with the Puppeteer, they could create all sorts of trouble for Ladybug and Chat Noir. They could play off each other's devious natures and come up with some really elaborate plans to deceive the heroes.
A really cool plot could involve the Puppeteer teaming up with other villains in the Miraculous Ladybug universe. They could form a sort of 'villain alliance' and plan a major attack on Paris. The heroes would then have to figure out how to break their alliance and defeat them. Also, perhaps the Puppeteer discovers a way to use the Miraculous powers for herself without having a Miraculous, which would be a huge threat to Ladybug and Chat Noir.
Ladybug and Chat Noir are obvious choices. Their dynamic is always interesting to explore further. You can show how they grow and develop as heroes. Also, the Puppeteer herself is a great character to focus on. Her motives and powers make her a complex character that can be delved into deeper in fanfiction.
One idea could be exploring the emotional aftermath for the characters after the events of Puppeteer 2. For example, how Ladybug and Chat Noir's relationship might have changed. Another could be introducing a new villain who takes advantage of the chaos left behind. Maybe a new Miraculous holder could also be introduced in the fanfiction, adding a new dynamic to the story.
In this fanfiction, it could be that the Puppeteer has allied herself with other villains. Together, they form a powerful group that is more than a match for the Miraculous holders. The heroes would then have to figure out not only how to defeat the Puppeteer again but also how to deal with her new allies. Maybe Ladybug and Chat Noir would need to recruit more help or find new Miraculous powers to combat this threat.
In an akumatized Ladybug fanfiction, the akuma might enhance her powers in unexpected ways. For example, her Lucky Charm could become a more powerful and unpredictable object. Her akumatized form might also have a darker and more menacing look, perhaps with a more aggressive fighting style. Writers could explore how the heroes and villains interact with this changed Ladybug.
Most likely, we'll see the Puppeteer as a central figure. Alongside her, Ladybug and Chat Noir are bound to be there as they always are to save Paris. Also, Alya might be involved as she often helps Ladybug. Since the Puppeteer has a history of using dolls, perhaps there could be a new character who is an expert in doll - making or some sort of antique doll collector who has some connection to the Puppeteer's powers or origin.
In this 'Miraculous Ladybug Criminal Ladybug Fanfiction', there could be a lot of potential storylines. One possibility is that an external force is influencing Ladybug to commit crimes. It could be a new villain who has found a way to manipulate the Miraculous. This would create a great conflict as Ladybug has always been the symbol of good. Her friends and allies would be in shock and would have to find a way to stop her without hurting her. The story could also explore her internal struggle, as she might be aware of what she's doing but unable to stop herself.