Well, think about it. Chiron could be a visiting counselor at the mental hospital in the fanfic. He tries to subtly guide Percy and the others without revealing their true nature as demigods. For the other characters, like Tyson, his large size and simple - mindedness might be mistaken for some kind of developmental disorder. But in reality, he has the strength of a cyclops. They all have to navigate this strange world of the mental hospital while still being true to their demigod selves, which makes for an interesting dynamic.
In this fanfic, the characters' demigod traits can be re - interpreted in the context of mental health. For example, Leo's ADHD - like energy and his ability to tinker with things could be seen as a hyperactivity disorder. He's always getting into trouble in the mental hospital because he can't resist fixing or building things. Piper's charm - speak could be misinterpreted as some sort of manipulative behavior disorder. And Percy, with his water powers, might accidentally cause things like faucets to explode when he's emotional, which is seen as a sign of his 'instability' in the mental hospital setting.
Annabeth could be in the mental hospital as a patient who is overly logical and her intelligence is misconstrued as obsessive - compulsive disorder. She tries to analyze everything in the hospital, from the routines of the staff to the behavior of the other patients. Grover could be there too, his fear and nervousness around humans being seen as a social anxiety disorder. Percy, as always, is the heart of the story, trying to figure out how to use his powers to protect his friends and also deal with his own 'diagnosed' problems.
Another possibility is that Percy has suffered some sort of mental trauma from his battles. He's having nightmares and flashbacks, and his friends think the mental hospital can help him. But once inside, he realizes that there are dark forces at work in the hospital. He has to use his demigod powers subtly to protect himself and the other patients from these unseen threats.
It's possible that the main antagonist is a new mythological creature that has the ability to manipulate minds. It has lured Percy and other demigods to the mental hospital and is slowly driving them insane in order to feed off their fear and chaos energy.
One possible plot could be that Percy wakes up in the mental hospital with no memory of how he got there. He starts to notice strange things like the nurses and doctors seem to have an otherworldly aura. As he tries to regain his memory, he discovers that the hospital is a front for some sort of mythological experiment. He then teams up with other demigods who are also trapped there to escape and expose the truth.
Percy Jackson would probably stand out in Forks. He has this whole demigod aura and his powers are quite flashy compared to the more low - key vampires and werewolves. But he might adapt by using his wits. He could blend in by pretending to be a normal human, but his constant awareness of possible threats (which in Forks are vampires and werewolves instead of Greek monsters) would make him seem a bit paranoid to the locals.
In a fanfiction, the Percy Jackson characters could fit into the Pokemon world in several ways. For example, they could be travelers from another dimension (the demigod world) who are trying to understand the rules and ways of the Pokemon world. Percy could form a special bond with water - type Pokemon, similar to how he has a connection with water in his own world. Annabeth could be intrigued by the architecture and layout of the Pokemon cities and use her knowledge to help the group navigate. Grover might get along well with Grass - type Pokemon as he has an affinity for nature. And they could all participate in Pokemon contests or battles, using their demigod skills to enhance their performance.
Well, he might meet other characters who also claim to have had strange experiences. They could form a sort of alliance, thinking they are all in the same boat, not realizing Percy's experiences are real in the context of the Percy Jackson universe. They could plan an escape together, believing they are being wrongly held.
One idea could be Percy starts having visions related to his demigod adventures in the mental hospital. The staff think it's part of his mental illness, but it's actually his connection to the Greek world trying to reach him. Maybe he meets other patients who are actually demigods in hiding too, and they have to figure out how to use their powers subtly without getting caught.
Perhaps a powerful monster in disguise as a doctor or staff member. They could be using the hospital's resources to create a portal to the underworld or something equally dangerous, and Percy has to stop them. They would be constantly trying to drive Percy crazy or keep him from interfering with their plans.
Well, Annabeth could use the Groundhog Day - like situation to study more about ancient prophecies. Since she's always eager to learn and solve mysteries, she would make the most of the repeated days to dig deeper into the knowledge she needs. Every day she might find new clues or make new connections that she couldn't before.
If we consider how the Percy Jackson characters would fit into LOTR's world, we also have to think about the different races in Middle - earth. The demigods might have different relationships with the hobbits, dwarves, elves, and men. The hobbits could be both curious and a bit scared of the demigods at first. The dwarves might be interested in the demigods' weapons and their fighting abilities. The elves, with their own magic and long - lives, could compare and contrast their existence with that of the demigods. And the men of Middle - earth would likely be both cautious and eager to use the demigods' powers for their own purposes in the fight against Sauron.