The more you look into this, the more complicated it gets. M-W's Dictionary of English Usage gives a tangled history of the words in which "run the gantlet" appears to be older than "run the gauntlet," and they say the notion that "gantlet" is more correct is "mistaken," and at any rate the words were never etymologically distinct. They also note that "British dictionaries never recognized the distinction, and "gantlet" has long since dropped out of use as a spelling variant in British English." What to do? Go with what people say or with what AP, et al., decree with very little solid justification? An excruciating dilemma!
Many style quibbles reach this point. We follow these rules not because of sound historical evidence, but because an editor somewhere (in this case, Norm Goldstein of the Associated Press) decided it was the best thing to do. He probably inherited it from someone else, who learned it in a high school classroom at the turn of the century.
Style doesn't necessarily have to be logical, or even right, to be style. It merely has to be consistent. Yes, gantlet and gauntlet have confusing histories. Yes, people may use them interchangeably. But the newspaper copy editor's bible, the AP Stylebook, has made its determination.
You make the call.