I'm seeing some evidence that Louisa and Caroline really weren't all that close, or at least that Caroline doesn't like Louisa's husband. Maybe Caroline's letter to Jane comes close to the truth when she says that two women can't spend the whole day together without a quarrel.

Caroline decides to live in her unmarried brother's house in the country, even though she would have more social opportunities at Mr. Hurst's house in town. When she and Louisa laugh over Elizabeth's muddy petticoat and wonder why she would go on such a long, dirty walk just because Jane was sick, Bingley's remark that it "shows an affection for her sister that is very pleasing" seems like a reproof. It appears that JA is setting up a contrast in the relationships of the two sets of sisters -- Jane and Elizabeth, who are truly close and would do anything for each other, and Caroline, who doesn't think a sick sister is worth getting one's petticoat muddy for.