OKay so far......
We have her trying to become her brother's educator which means she is in a competition against her little brother's uncle.
They both need to curry favor towards nobles in order to win the vote, (although the little brother's uncle might be overconfident and not plan accordingly although this part is not known just my prediction based on his character) and in order to do that, the princess in name only is going to use hella money to better the slums condition.
She's holding a banquet/ball in order to do so but why? IDK that part's foggy, I think it's just to feed into the noble's superficial need for pride and honor.
All the while she still hasn't given a formal reply to Lancelot's close friend (the ML, I forgot his name) after he proposed to her in order to have her be fully eligible to apply to be her little brother's teacher. I don't know why, foolishness I suppose.
Her little brother, on the other hand, doesn't want to let her go but is using the excuse of her not being good enough for Lancelot's best friend (although there's probably truth in this as well.... OOp) and princess in name only has confused it for a disorder because despite being a sibling in both of her lives psychology just isn't something she understands....
Did I get it all?