In terms of her relationship with the Cullen family as a whole, they would be quite concerned. Carlisle, for instance, might see Bella's new fascination as a dangerous path. Esme might try to mother her more to bring her back to her old self. The other Cullen siblings could have mixed reactions. Alice might see the future implications and try to guide Bella away from this new path, while Jasper might be more on edge around the new, more unpredictable Bella.
Her relationship with Edward would definitely change. Edward is used to a more innocent and reserved Bella. If she becomes a Katherine Pierce fan, she might become more assertive and independent, which could either make Edward more attracted to her in a new way or cause him to be wary. He might not like the new, more dangerous side of Bella that is emerging.
With Edward, their relationship would be different. They'd be more equal in terms of strength and power. No longer would Edward need to constantly protect Bella. Instead, they could fight side by side against their enemies. Their relationship might also have a new dynamic of competition, as they both explore their new vampiric abilities.
Maybe Bella begins to idolize Katherine so much that she tries to emulate her style and behavior. This could lead to a rift between her and Edward, as he doesn't understand her newfound fascination. Bella might then get involved with some of Katherine's old enemies or allies, unknowingly bringing a whole new set of problems to Forks. It could also involve her trying to learn some of Katherine's vampire - related skills, like compulsion, if that was possible in the context of the story.
When Bella has a power like super strength in fanfiction, it can change how others view her. Her friends might be in awe of her new ability, but it could also make some of them feel inferior or intimidated. Her relationship with Edward might also change as he has to adjust to not always being the one to protect her. He might feel a bit of jealousy or pride in her newfound strength.
She might become more assertive. In the Twilight series, Bella is often rather passive, but in the Mikaelson family, she would need to be more forceful to hold her own. She'd have to learn to be more cunning and strategic, like the Mikaelsons are.
She would likely become more confident in her own strangeness. Instead of being the somewhat shy and self - deprecating Bella we know from the Twilight series, she would embrace her darker side. For example, she might not be as hesitant to stand up to those who try to bully or intimidate her.
If Bella becomes a banshee, her relationship with Edward would be very different. He would be more protective of her as she has a new and potentially dangerous power. He might also be a bit wary as banshees are known for their wailing which could attract unwanted attention. With Jacob, he might see her as even more of an enigma and be more drawn to her in a way, since she's now a more powerful supernatural being.
At first, their relationship might become strained. Bella's sister would have to keep her new identity a secret from Bella, which could create distance. But once Bella finds out, they could become closer as Bella tries to understand what her sister is going through. Bella might be more protective of her sister now that she's in a more dangerous world.
In fanfiction where Bella Swan becomes a Pierce Vampire, her appearance might be different. Pierce Vampires could have distinct physical features. Maybe her eyes turn a different color, or she has a more ethereal glow. Her style of dressing could also change to reflect her new vampiric status. She might be more drawn to darker, more elegant clothing. Her interactions with other characters would be fascinating to explore. How would the Volturi react to her transformation? Would they see her as a threat or an ally? And how would her human friends, like Jessica and Angela, perceive her now that she's a different kind of vampire?
Her relationship with Edward would likely become more equal. As a human, Bella was often in a more vulnerable position, but as a vampire who has flipped her switch, she could stand on equal footing with him. They might engage in more intense and equal - sided battles or hunts together.
The story would change drastically. For starters, her relationship with Edward would be very different. Instead of being the vulnerable human he has to protect, she would be a powerful being in her own right. Their dynamic might shift from protector - protected to more of an equal partnership, or even a power struggle if their goals don't align.