This literally means nothing. This is just an annoying gnat flying around not knowing its unwanted. The cousin has negative chance of being a fiancé no matter what. She's already been seen completely through and the prince has made it clear to everyone that he doesn't want to marry especially her. She can whine, but nothing she does against the MC will change that. The challenge is dumb. The Prince as already made it clear, as the maids said, that he's smitten with the MC and very not smitten with his cousin. Win or lose the Prince will continue to pursue the MC and neither party is under any obligation to do what the other said, nor is there any reason for them to. The cousin SHOULD move on from the prince and pursue her future otherwise, but if she doesn't currently want to then there's no reason for losing a tea party challenge to change that. And again the MC is the one being chased, not the other way around. She's already turned down the prince's engagement. Maybe she should say that to the cousin, "I've already turned down the prince's engagement. I am only here in the royal palace because he insisted on an opportunity to court me... and I guess I want to raise my self confidence or whatever."
This story has no real reason to exist and it keeps taking away any semblance of conflict and story telling it kinda has. The first fiancé is evil...but actually no, he's just an idiot. Feel bad I guess. The family is evil but they never get any screen time. This fiancé rival is the new antagonist, but no. The prince shut her down several times already. This story could honestly be over with already. She isn't having to fight her first fiancé to get out of the marriage. She isn't have to fight her family. The prince is already deeply in love with her and they're only not engaged because she has no self confidence. This could have been a story of, "If I manage to do XYZ on my own power, then we'll get married." that'd be interesting.