I know the story--the claim was that the person had thought-traveled to Mars, lived with the martians, yada yada. As you say, the key to discrediting the fraud is that the Earth is closer to a double planet than not because of the Moon.
It is driving me up a wall that I can't find the name of the story. I own almost all of Asimov's SF stories and most of the novels. "Own" but do not "have" due to loans. There were a number of books where he paired a science essay with a SF story on the same theme. I vaguely remember an essay on planetary albedos that may have gone with it.
If you think of it or come across it I'd love the title.
