There could be a new character introduced. This character could be someone from Bella's past or a completely new entity. Maybe this character has a claim on the person Bella loves, and Bella, being the selfless person she is in this fanfiction, steps aside to let this new character have a chance at love.
One idea could be that Bella steps aside because she realizes her love is causing more harm than good to the people around her. Maybe she sees that her presence is putting her friends in danger, so she decides to leave for their safety.
There could be Bella's partner in the relationship she is betraying. This character might be a loving and trusting person, which would make Bella's affair even more complicated and morally wrong.
Often, you'll see Edward in these fanfictions. Even though the story is about Bella as a vampire before Twilight, Edward still plays a role. Maybe he's a mystery to Bella at first, or he's from a different vampire group that she has to interact with. Another common character is Jacob. He could be in a more antagonistic role, being jealous of Bella's new vampire life. And of course, the Cullens might be mentioned or even play a part. Carlisle could be a figure that Bella looks up to for advice on being a 'good' vampire.
There could be Edward. Even though the focus is on Carlisle and Bella, Edward's presence might still be felt as he was an important part of Bella's life before. His reactions to Carlisle and Bella's relationship could add some drama.
There could be classic creepypasta characters like Slenderman. He is a well - known figure in the creepypasta universe, often associated with mystery and fear. Another possible character could be Jeff the Killer. His creepy appearance and murderous nature would add a lot of horror to the fanfic. There might also be characters like Eyeless Jack, who has his own unique and disturbing backstory.
Alice could be a great addition. She could be Bella's best friend who is always trying to play matchmaker between Bella and Jasper. She has this uncanny ability to sense when things are going well or not so well between them and gives her two cents whether they want it or not.
Edward, of course. He's Bella's main love interest in the Twilight series, so it's very likely he'll be by her side in the hospital. He would be the one to comfort her, and his vampire abilities might come in handy, like his super - speed to get things she needs quickly.
Alice is often a very popular character. Her ability to see the future can add an interesting element to the wedding planning. For instance, she might foresee some problem with the wedding dress or the weather on the big day and try to prevent it. Jacob also makes an appearance in some fanfictions. His presence can create some tension as he still has feelings for Bella.
Since I haven't read the fanfiction, I can't say for sure. But it's possible that there could be new characters introduced to either support or oppose this strange relationship between Bella as Jasper's pet. Or perhaps some existing characters from the original story might have different roles or personalities to adapt to this new situation.
Alice could be a great addition. She could use her ability to foresee events to help Edward and Bella avoid getting caught by the police. Another good character would be Carlisle. He could be the voice of reason, trying to convince Edward to turn away from his criminal ways. Jasper might also be included, perhaps using his power to influence emotions to help in certain situations.