Ma’ah’s brother says all the “I can see the original Kahless” stuff at the end of the episode. That means that Kahless II being a clone is common knowledge. I wonder when that happened, though Kor seemed aware of it in DS9:”The Sword of Kahless”.
Worf insists that the truth about the cloned Kahless be revealed to the people at the end of "Rightful Heir", so I assume that info was part of the press release when Kahless was crowned.
I've always liked the idea that the Klingons are passionate enough about Kahless' promise to return that they're completely satisfied that the clone fulfils the prophecy.