One great 'Harry and Charlie Swan Fanfiction' is 'The Unlikely Bond'. It beautifully weaves their characters together in an alternate universe where they team up to solve a mystery in Forks. The story has a great balance of character development and plot twists.
The popularity of 'Harry and Charlie Swan Fanfiction' lies in the interesting combination of two different universes. Harry comes from the magical world of Hogwarts and Charlie from the more mundane but still exciting world of Forks. When combined, it creates a lot of opportunities for unique stories.
There could be a story where Charlie Swan adopts a magical creature that has escaped from the wizarding world. He doesn't know it's magical at first but then realizes it when strange things start to happen around his house. He then has to figure out how to return the creature to its proper place in the wizarding world without getting into trouble with the Ministry of Magic.
Charlie Swan can be integrated as a wizard who has chosen to live in the Muggle world. He could have his own set of magical abilities that are useful in both worlds. For example, he might be really good at spells that can calm wild animals, which would be useful in his job as a sheriff in Forks and also in the wizarding world when dealing with magical beasts.
Often, the theme of dealing with the supernatural is present. Charlie has to come to terms with the vampires and werewolves in his town. Also, the theme of loneliness can be found, as he is a single father trying to raise his daughter in a strange situation.
I'm not aware of it being a particularly popular fanfiction. There are so many 'Twilight' fanfictions out there, and ones that deviate so far from the original story's relationships may not gain mainstream popularity.
One interesting 'Harry Potter Charlie Swan' fanfiction could be one where Charlie Swan discovers the wizarding world of Harry Potter. Maybe he stumbles upon a magical creature in the forests near Forks and that leads him to Hogwarts. It could be a story full of adventure as he tries to understand this new world while still maintaining his connection to his life in Forks.
The Volturi could be a main antagonist. They might see the intrusion of the wizarding world into their territory as a threat to their power and secrecy. They could try to eliminate Harry Potter and Charlie Swan to protect their vampire - centric world in Forks.
I'm not sure specifically as it's a fanfiction. Fanfictions can have a wide variety of interpretations and storylines. It might explore an alternate scenario where Charlie Swan and Bella are in a relationship that leads to such a situation, but this goes against the original 'Twilight' story's intended relationships.
Another place to check could be some dedicated Harry Potter or Twilight fan forums. Sometimes, fans share their own written fanfictions on these forums. You might have to dig a bit deeper, but it could be worth it. There could be some really unique and interesting 'Harry Potter Charlie Swan' fanfictions that haven't made it to the bigger platforms yet.