...that perhaps the Bingley sisters get along less well in private than they do in public, aren't there? ;-)

However, as far as Caroline choosing to join her brother in the country goes, we are also told that Mrs. Hurst is still glad to act as though her brother's house were her own (perhaps for money reasons?), and Caroline as a young single woman would always need a chaperone. Unless she was staying with friends, she couldn't stay in London on her own. Plus, she might be glad to be the one in charge of the household for a change!