Here's at least some evidence that the King knew he was not completely invincible:
Quote:
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"That this [The Shire] would prove to be not far from the hated land of Lorien seemed to him [the Lord of the Nazgul] not unlikely, if it was not indeed within the fences of Galadriel. But the power of the White Ring he would not defy, nor enter yet into Lorien."
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He seemed to fear Galadriel. One might argue that it was Nenya he feared, but was not the location of the Three secret to Sauron, though he had suspicions? The wording here almost implies that the King knew of her ring, but it doesn't necessarily. Only that he would not defy the fence about Lorien -- this being the power of the White Ring.
I don't think he would've quoted his prophecy had he known he was up against a woman. He probably would've laughed at her, but I doubt he'd have busted out the "no living man" thing. I suspect he knew the prophecy wasn't all-inclusive. Any arguments to the contrary?