Well, at first, the Cullens might be more suspicious of Bella since she's related to James. But as they get to know her better, they might see that she's different from her brother.
Since Bella is James' sister, the Cullens could be more guarded around her. However, Bella might try to prove herself to them, which could lead to a slow - building trust. She might have to work harder to be accepted into their family compared to the original story. For example, she could offer to help them with things related to vampire lore that she learned from her family, including James. This would gradually change their relationship from one of suspicion to a more friendly or even family - like bond.
In this fanfiction, the relationship between Bella and the Cullens would be completely different. The Cullens would be very cautious of her at the start. But as time goes on, Bella could show that she has different values than James. Maybe she helps them in a time of need against a common enemy. This act of selflessness could start to change their perception of her, and over time, they could develop a deep respect and even love for her, just in a much more complex and different way than in the original Twilight story.