Can someone help me with pinning down a specific conflict in "Bernice Bobs Her Hair" that was fueled by childhood in some aspect?
Fitzgerald does not necessarily delve into the respective childhood periods of the main characters. Part of this is reflective of the time period, something of which Fitzgerald was extremely aware. The 1920s was a time period rooted in the now, in the eternal present, and in a belief that temporal reality transcended everything else. It is for this reason why "bobbing" one's hair carries so much importance at the time. It was an expression of the present tense. Little else of the time period mattered.
Certainly, one can project as to examining childhood paradigms that influence the conflicts characters face. One such conflict would be the relationship between Marjorie and Bernice. The former experienced greater social acceptance and carried herself with more panache and style than the latter. For her part, Bernice was seen as priggish and more of a "prude." She was seen with social skepticism, more likely to be marginalized than anything else. As the narrative opens, Marjorie is exasperated with trying to "set up" her cousin, someone that she views as a tether to her social advancement. The way in which both characters relate to one another in childhood or at least in a younger capacity, where Marjorie was on the top of the social hierarchy and Bernice underneath her, is inverted as the narrative progresses. The ensuing conflict is rooted in childhood in that the "natural order" was flipped with Bernice becoming popular. One can see how the central conflict of the story is rooted in childhood.