To get Caroline Bingley's character right, you have to understand that she is a foil for both the Bennet sisters and Mr. Darcy.
She is the more rational choice for Darcy vs. the Bennets. She has education, manners, a fortune, and clearly, relatives that he likes. So many fan fiction authors make her vulgar and/or unfashionable, but she isn't! That is why Darcy enjoys hanging out with her in the beginning; he would not have her at his house if she was embarrassing. Even when angry with Elizabeth, Caroline does not dare go further in attacking Elizabeth at Pemberley. She has self control. She understands boundaries, which Jane and Elizabeth mostly do, but the rest of the Bennet family struggles with. This is why she's a foil for them.
As for Darcy, at the beginning, Caroline is a nearly perfect mirror of his opinions and snobby attitude. She is doing this on purpose as a way of flirting, but it's probably pretty close to her real personality anyway. She's right that Darcy looks down on Elizabeth's uncle being a lower class lawyer. She's right that he finds the Bennet family intolerable to marry into. However, as Darcy falls in love with Elizabeth and then reforms, Caroline's mirror distorts. That shows his growth in the novel. She, like Elizabeth, fails to update her priors, though to be fair to Caroline, she didn't build her knowledge of Mr. Darcy on first impressions. It's harder to change her mind because she did once know him very well.
Side note: this is also why people woobyfying Darcy hurts Caroline as a rational character. They start in a very similar place and love mean girl gossiping together, then he changes. When Darcy's flaws are erased, it makes Caroline look super irrational and much crueller.
Lastly, Caroline is above all else, pragmatic and strategic. She does not hold grudges once it becomes more advantageous to drop them. She would never, ever, now that she is connected by marriage to the Bennets, mock them in public. Because that reflects on her! Caroline would be in London talking up that the Bennets are a very old gentry family with an ancient estate or something. She's going to be giving them a PR makeover to all her fancy friends because they are HERS now, for better or for worse and whether she likes any of them or not. Yes, in private she might be mean, as she is in the novel, but again, she's not vulgar and she has nothing to gain in public. She has manners, she has self-control; being a mean girl doesn't override that.
