As it is the O.N speaking, I'm also inclined to think Miss Bingley is not desperate. She does not even appear depressed during the visit. While she still feels jealous of Darcy's admiration of Miss Eliza's 'fine eyes' while he never noticed her (if he ever had Caroline would tell us!) she knows that Darcy wasn't contemplating an attachment when he made that remark. It would stand to her reason if Darcy did not wish her brother to marry Jane on account of her 'low descent' then Darcy is certainly not going to marry Lizzy. She would not know of his proposal to Lizzy and she doesn't realize Darcy is in love with Lizzy. So how could she should be desperate? I think it likely Caroline has some moderate hope. She may accrue up her accreditations- Accomplishments, Acquitance with Georgiana, Dowry, Pemberley visits, Friendship with brother Charles. But none of these, not even the latter can make Darcy more attainable for her as he does not love her. Her 'connections' are not great. Caroline's family fortune is based on trade, there's no family estate, she has no gentry relations. Lizzy is at least the daughter of landed gentry so has a higher social status than Caroline. However, Caroline would not comprehend this. Yet, she would still see no danger as she knows Darcy discouraged Bingley's attachment to Jane. Caroline has little understanding of Darcy who is a complex man.