There was no question Teapot Dome would have surfaced: Harding knew in 1923 he had been betrayed by a number of his appointees and friends. It's not out of the question that he might have asked to address a joint session of Congress, during which he would have laid bare Teapot Dome, naming names and relieving those men of their duties to prepare their courtroom defenses. Further, it's not out of the question that this version of Harding might well resign the presidency,saying to the nation that he had let the nation down and that he was ultimately responsible.
That's one way Harding lives, albeit in retirement / exile, until 1925. Calvin Coolidge would have no problem here winning a term in his own right: as Mikestone pointed out, the Dems were in sorry shape in the '20s, and wouldn't be able to provide a true consensus candidate, never mind someone who could defeat an incumbent. About the only effect Teapot Dome might have would be the reduction of Coolidge's margin of victory.