Bostock is irrelevant. Your friend can sue under Burlington v. News Corp.
Burlington answered this exact question ten years ago, using exactly the same, well-established logic used in Bostock.
Burlington involved a news anchor (Burlington) who was fired after using the N-word descriptively in a staff meeting. He sued under Title VII. He argued he was being discriminated against on racial grounds, since several Black employees had used the word at work without consequence.
The case was tried before a jury in Federal district court. Burlington lost after the jury decided the facts did not support his claim. However, the judge made clear that the actions he alleged would violate Title VII:
Historically, African Americans' use of the word has been ironic, satirical, or even affectionate. Too often, however, the word has been used by whites as a tool to belittle, oppress, or dehumanize African Americans. When viewed in its historical context, one can see how people in general, and African Americans in particular, might react differently when a white person uses the word than if an African American uses it.
Nevertheless, we are unable to conclude that this is a justifiable reason for permitting the Station to draw race-based distinctions between employees. It is no answer to say that we are interpreting Title VII in accord with prevailing social norms. Title VII was enacted to counter social norms that supported widespread discrimination against African Americans. To conclude that the Station may act in accordance with the social norm that it is permissible for African Americans to use the word but not whites would require a determination that this is a "good" race-based social norm that justifies a departure from the text of Title VII. Neither the text of Title VII, the legislative history, nor the caselaw permits such a departure from Title VII's command that employers refrain from "discriminat[ing] against any individual...because of such individual's race."
Added: Why Bostock is Irrelevant for Your Friend’s Case:
As you say, if we substitute the word “race” for “sex” in those sentences from Bostock, it seems obvious your friend would have a case under Title VII. This is true, but not because of Bostock. Bostock changed who could claim sex discrimination under Title VII, but nothing else. In particular, it did not change the test used to prove discrimination under Title VII. The sentences you quoted, use the well-established ”but for” test.
Here's Gorsuch’s explanation of how the "but for" test works:
In the language of law, Title VII’s “because of” test incorporates the “simple” and “traditional” standard of but-for causation. That form of causation is established whenever a particular outcome would not have happened “but for” the purported cause. In other words, a but-for test directs us to change one thing at a time and see if the outcome changes. If it does, we have found a but-for cause.
The Bottom Line: Bostock is illustrative, but not dispositive, of your friend's claim.