Why does Hamlet not kill Claudius when he has the perfect time and opportunity?

This a moment of dramatic irony in the play, meaning the audience has information which Hamlet does not. The audience is able to overhear Claudius at his prayers. The audience knows that Claudius does not feel remorse nor does he repent of killing the senior Hamlet and taking the crown. Claudius is on his knees expressing his inability to engage in confession of his sins. He is not in a state of grace.
Hamlet comes upon him kneeling at prayer and looking for all the world as if he is in a state of grace. At this point, having staged his mousetrap play, Hamlet knows that the ghost's story about Claudius is true. This would be the perfect moment to act: Claudius is alone and defenseless, unaware of Hamlet's presence. Hamlet, however, looking at the external appearance of Claudius, believes he would not exact justice if he killed his uncle at that moment. Claudius, he thinks, would go to heaven while his father, who never had a chance to repent of his sins, would walk the earth as a ghost. That seems unfair—not true revenge—so Hamlet waits.

We've reached act 3, scene 3 and Claudius is down on his knees in prayer. He opens his heart to God and expresses his sense of guilt at having murdered his brother, Hamlet's father. All murder is bad, but there's something particularly depraved about killing your own brother:

Oh, my offence is rank. It smells to heaven.
It hath the primal eldest curse upon ’t,
A brother’s murder.

With Claudius all alone, and in such a prone position, he is vulnerable to attack. Just then, Hamlet slips gingerly into the room, observing Claudius from a safe distance. Now would be the perfect time for him to exact revenge for the death of his father and kill Claudius while he's still in prayer. Hamlet starts to remove his sword from the sheath but then hesitates before putting it back.
 
So why does Hamlet miss his big opportunity to strike? He tells himself that if he kills Claudius while he is at prayer then there is a risk that Claudius will go to heaven, not hell. That is too big a risk for Hamlet to take. Far better, Hamlet reasons, to kill Claudius when he's engaged in some sinful act such as getting drunk, swearing at the gaming table, or sleeping with Gertrude.