First off, the four year old kid in her will notice something Archie is at pains to deny: he's something called a magnetic hero. He doesn't like having the idea of having an advantage but there's something about him that draws most people towards him despite being otherwise incredibly average. Simply put, the whole thing will start with the little kid in her yelling "WANT THAT!!!" (This part of her dislikes being lectured to by some jerk in a sweatshirt that she can't just grab everything she wants.)
Second, we've gotta deal with the eight year old kid in her that hates the idea of being embarrassed. Coming in second place to the first girl is going to make her look bad and she'll do anything she can to avoid it. (This part of her psyche is also the one that gets pissed off when some unpleasant bohemian tells her that she's a selfish, thoughtless and shortsighted jerk when she gets her bowels in an uproar when she should know going in that Archie can't really afford to treat her the way she's accustomed to. The part of her that agrees with Jughead is not as strong as the part that hates to look foolish.)
Third, we have to deal with the bratty twelve year old that can't stand the presence of a rival because she's too used to getting her own way and also thinks that she's more or less the victim of a plot to make her look like some rich jerk waltzing into a situation and yelling "King Me" because her daddy is loaded. Having some burger-eating clod tell her that there's a place she shouldn't be and a relationship she shouldn't step into will not go down well.
Finally, we have the core identity that's starting to emerge: someone who, despite reluctance to mingle with the employees' children because for some odd reason, Daddy tells her that she's becoming something he doesn't like, is enthralled by the new environment in which she finds herself and wants to become part of it. Granted, that part is alpha female but quite frankly, the wrench wench with the poor fashion sense shouldn't take things so personally. Now, if only Daddy would stop spouting the same silly noise about becoming a spoiled, pampered and selfish heiress that Archie's creepy friend does, her life in Riverdale would be almost ideal.