If the intermarriage from the family story occurred in Oklahoma, it can still work - the marriage in question might have been with someone who was themselves mixed race. Many of the Cherokee were themselves of mixed race generations before the expulsion to Oklahoma.
Remember, we are looking at the marriage of Warren's parents, nearly a century ago. The "one drop" rule was still a thing, in culture at least. So Warren's mother's family was suspected of being part native American. that could easily mean that they just knew that Warren's mother's grandmother married a Cherokee who might have had predominantly white ancestry to begin with
(I know that legally, the "one drop rule" didn't apply to Native American ancestry. But culturally, it probably still did in many circles.)
I gotta add, one of the whitest white people I've ever known (fair skin, blue eyes, white-blonde hair) had an ancestor on the Dawes role and was an active member of the Cherokee tribe. You can't miss those strong twangy Okie accents. You don't need to look even remotely Native American to actually be Native American, in the sense of having shared ancestry and culture.