Perhaps a new, powerful demigod - vampire hybrid that Percy has to face. This hybrid could be causing trouble in Forks, and Percy has to stop it before it destroys the delicate balance between the vampires, werewolves and humans there.
It could also be a group of angry werewolves who see Percy as an intruder. They might think that Percy's presence will disrupt their relationship with the vampires and put the safety of Forks at risk, so they try to drive him out.
A powerful sorcerer could be the main antagonist. He might be after Percy's powers or trying to use the school as a base to spread chaos. Maybe he wants to open a portal to the underworld right in the middle of the schoolyard.
Maybe it's a corrupted version of one of the gods from Percy's world who has gained even more power in this new dimension. They could be trying to spread chaos and destruction, and Percy has to stop them. This antagonist would know Percy's weaknesses from their time in the original world, which would make the conflict more intense.
It could be a new teacher who is secretly a minor villain from the Percy Jackson universe. This teacher might be trying to undermine Percy's confidence and expose his demigod side to the mortal world. They could set up difficult situations for Percy in class, like impossible tests or false accusations of cheating. Also, they might try to turn Percy's friends against him by spreading rumors. The teacher could be working for a bigger bad guy in the background, but their main goal is to make Percy's high school life miserable.
There could be an ancient and evil force that has infiltrated the demigod world. This force uses Percy as a scapegoat and arranges for him to be sent to prison. It wants to cause chaos and destruction within the demigod community. It manipulates the situation from the shadows, and Percy has to figure out its identity and motives while in prison in order to stop it.
In a fanfiction where Percy Jackson goes to Forks, he might be really confused by the gloomy weather. He's used to more sunny and open places. He could meet the Cullens and there would be a clash of his demigod powers and their vampire abilities. Maybe he'd mistake them for some new kind of monster at first.
It could be a dark wizard who has infiltrated Hogwarts under a disguise. Percy might be the one to first suspect this wizard's true identity.
Another possibility is a demigod from the past who is jealous of Percy's knowledge and power. This demigod could be someone who was once in a position of power or influence but feels threatened by Percy's sudden appearance and her attempts to change things. They might try to undermine her at every turn, whether by spreading false rumors about her or directly attacking her when she least expects it.
It could be Tartarus itself personified. As Percy is alone in this terrifying place, Tartarus could take on a form and directly oppose him. It would use all the horrors and challenges of the place to try and break Percy.
It could be a dark wizard who has also discovered the secrets of Forks and wants to use the town's unique location or some hidden power there for evil purposes. This dark wizard might see Harry and Ron as threats and try to get rid of them. He could be using dark spells to manipulate the townspeople or cause chaos in Forks.
Another option could be a group of rogue demigods who have turned to the dark side. They follow Percy to Forks, hoping to find some powerful artifacts there. They don't care about the vampires or werewolves and are willing to cause destruction to get what they want. Percy has to stop them while also dealing with the local supernatural creatures.