I bet half of the house girls, including aforementioned Lucy don't enjoy working in the club. But that kind of job sucks in, I guess. Easy money and quite good one. For example, the new girl Nicolette. The side character's guide literally says about her: "This poor girl’s showing you what’s gonna happen to Rosalind if she doesn’t win the competition".
Rosie doesn't strike me as a particularly strong willed or very intelligent woman. She is kind and has good soul but that's it. Of all three Carnations she is the most likely to make it into the house girls and get stuck there until there is the demand for her. Even if the debt is paid, she doesn't have good job and has a child to care for. Pretty much like the MC's mom.
As I mentioned before, the least likely candidate to remain in the club after the Exhibition is Veronica. She is stubborn, determined and most importantly lesbian