If her environment was such that white people felt different, or felt like outsiders, why not allow them to make a club? The "white advnatage" may not exist in her school. In a social sense, in more liberal or urban environments, the "white advantage" is rapidly turning into a burden or negative connotation, regardless of the economic reality of racial demographics.
Yes, white people are at the top of the economic power structure, but holding that against every white person would be equivalent to telling black people because many of their brethren are disenchanted and starving they don't have the right to leave the projects.
Your intentions of racial equity are good, however in your opposition to this they contain a contradiction: regardless of the "white advantage" that is widespread, this girl most likely feels a racial disadvantage. Whether unintentionally or intentionally she likely feels racially disadvantaged, and the "white advantage" doesn't apply in her situation, even if her distant uncle is a CEO or some representative of the white power structure. Thus, denial of this would be in opposition to many ideas you likely support in favor of equal minority status.