Another possibility is that there could be a dragon hunter in the fanfiction. This person might see dragons as nothing more than trophies or sources of wealth. They would try to stop Hermione from moving the dragon away so that they can capture it themselves. They would likely be very skilled in tracking and capturing magical creatures and pose a real threat to Hermione and the dragon.
Maybe the Ministry of Magic could be the antagonist. They have their own rules about magical creatures, and they might see Hermione moving away with the dragon as a violation. They could send Aurors to stop her, believing that she's putting the wizarding world at risk by taking a potentially dangerous creature away. Even if Hermione has good intentions, the Ministry might not understand or care.
The main antagonist could be some dark wizards who want to use the dragon for their evil purposes. They might be after the dragon's scales, which are valuable in making powerful dark magic items, or its fire, which can be harnessed for destruction. They would oppose Hermione's attempt to move the dragon away as it would thwart their plans.
One possibility could be that Hermione moves away to protect the dragon. Maybe she discovers some dark forces are after the dragon for its magical properties, and by moving away with it, she hopes to keep it safe in a secluded location. She might have to face many challenges during this journey, like finding a suitable place for the dragon and dealing with those who oppose her decision.
Obviously Percy would be one of the main characters. And there could be a new friend he makes in the new place, let's say a girl named Lily who helps him explore the new town.
Perhaps a powerful magical being from a different realm is the antagonist. This being might mistake Hermione for someone else or see her as a key to something in the wizarding world. So, it uses its magic to make her disappear. This being would be very difficult to defeat or negotiate with, making it a huge challenge for those trying to find Hermione.
Maybe a new character could be the main antagonist. A powerful and mysterious being that Luigi stumbles upon during his escape. This character might have some vendetta against the Mario brothers or just be generally evil and wants to stop Luigi's journey for its own malicious reasons.
It could be Voldemort. Even though he's from the Harry Potter universe, if he finds out that Harry is hiding in Forks, he could send his Death Eaters there. This would not only put Harry in danger but also the Cullens and the people of Forks as the Death Eaters might start causing chaos.
There could be new characters introduced in this fanfiction. For example, a local wizard in the place where Harry moves to could become a main character. This wizard could initially be suspicious of Harry but later become his friend. Also, a magical creature that Harry befriends in his new location could be a central figure. It could help Harry adapt to his new surroundings and teach him things about the local magic that he didn't know before.
One possible reason could be that Hermione wants to pursue her own independent magical research. She is highly intelligent and may feel the need to focus on her studies without the distractions that come with being close to Harry. Another reason might be a disagreement over some major decision in the wizarding world. For example, if Harry has a different stance on how to deal with the remaining Death Eaters or how to reform Hogwarts' curriculum, Hermione, with her strong opinions, may choose to distance herself to avoid constant conflict.
It could be an old muggle acquaintance who is jealous of Hermione's success in the wizarding world. Maybe they sense that she's different and are determined to find out her secrets and expose her.
It could be the same group or person who tortured her in the first place. They might be trying to finish the job or silence her before she can expose their evil deeds.
Hades would be a likely main antagonist. He rules the Underworld and might see Hermione as a threat or an opportunity. If he thinks she has something valuable, like her knowledge of magic, he could try to keep her trapped in the Underworld. Or he might oppose her if she's trying to change something in his domain that he doesn't want changed.