Perhaps a storyline where Bella, as Jane's sister, has some hidden powers that are awakened due to her connection to the Volturi. This could make her a target for other vampire factions. And she has to rely on both the Cullens and the Volturi (including her sister Jane) to protect herself. At the same time, her relationship with the other Twilight characters like Jacob would also change as she becomes more involved with the vampire politics because of her sister - related to the Volturi.
One possible storyline could be that Bella discovers her sisterhood with Jane and tries to use it to protect the Cullens from the Volturi. She might try to negotiate with Jane, using their family bond to sway her decisions regarding the Cullens.
Yes, in some fanfictions, creative authors might rewrite the storylines to make Bella Swan be Jane's sister. It allows for a whole new set of plot developments, relationships, and character interactions.
One possible storyline could be that Bella discovers her Winchester heritage and has to learn how to hunt supernatural creatures like the rest of the family. She might struggle at first, but with the help of Sam and Dean, she becomes a skilled hunter.
A really interesting storyline could be Bella trying to reconcile her old life in Forks with her new life as a Mikaelson. She might struggle with the differences in values and lifestyles. For example, the Mikaelsons are more into power plays and manipulation in the supernatural world, while Bella was used to the more straightforward (although still complex) relationships in 'Twilight'. This internal conflict could be a major part of the story, along with her trying to find her place within the Mikaelson family and the wider supernatural community.
One possible storyline could be that Bella discovers her true heritage as Rowena's daughter and starts to learn magic. This could completely change her relationship with the vampires in the Twilight world as she now has a new power to protect herself.
One possible storyline could be Bella's struggle to fit in with the Pierce Vampire community. She might face discrimination or challenges because she was originally a human associated with the Cullens. Another storyline could be her search for her true identity as a Pierce Vampire. Maybe she discovers she has a special role or prophecy associated with her new vampiric form. And she could also be involved in a power struggle within the Pierce Vampire hierarchy.
A possible plotline could be that the Winchester sisters are magically transformed into Bella Swan right before a major event in the 'Twilight' series. They then have to quickly adapt to her life and relationships while also trying to figure out a way to get back to their own world. They might use their combat skills to protect themselves from the vampires and werewolves in a more aggressive way than Bella usually does.
One possibility could be a story where Bella Swan's sister discovers her connection with Jane in a world filled with vampire politics. Their relationship might start off rocky as they come from different 'sides' but gradually they find common ground.
One possible storyline could be that they discover a family secret together. Maybe something about their parents or their family heritage that has been hidden from them. This discovery could lead them on a journey to uncover more about their family's past.
A possible plotline could be that Bella's sister is initially in danger, and Carlisle saves her. This event sparks their connection. Then, they start to secretly meet, trying to hide their relationship from both the human world and the Cullen family. Another plotline might be that Bella's sister has a special power that attracts Carlisle, and they explore their relationship while trying to figure out what this power means for them.
I'm not sure of a specific one off - hand. But you can start by looking on popular fanfiction websites like Archive of Our Own. Just search for 'Bella Swan sister Jane mate' and you'll likely find a variety of stories with different takes on the concept.