Right, so, by Dumbledore's thinking, if he shows up having beaten Voldemort then he's not real. He expects that Voldemort would kill him. So the mirror could be a trap for any "triumphant Dumbledore" who appeared before it.

This is particularly true because Dumbledore doesn't fear death. He could easily have made himself forget the trap. If he wins and is killed by his trap, what's the harm? Its just the next great adventure.

To actually get the Stone probably just needs some terminal values equivalent of the arbitrary physical description Harry gave. A crazy person Dumbledore happens to know who thinks they are the only person in the world. Someone who doesn't know anything about the mirror at all. The possibilities are endless. Flamel himself is the obvious choice, identified as a person who knows some bit of trivia (s)he learned in 600 years alive.