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I just looked it up, and apparently there is only an 11.67% chance of an A flopping without a K.

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This can't be right. Probability that an Ace flops given that you have KK is 1-[(46/50)(45/49)(44/48)]= 0.225. I can't see this number being divided in half by including the times you flop a set.

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Oops. You're right.
P(overcard)=22.55%
P(no set)= (48/50)(47/49)(46/48)=88.24%
P(overcard and no set)=(.2255)(.8824)=19.9%

That's what I get for trusting Ken Warren's calculations.

19.9% chance of flopping an ace and no set when I have KK is uncomfortably high to put in >40% of my stack and then fold if he bets. So then maybe shoving preflop is best after all...or commit to the pot preflop but put the money in in two steps, regardless of what flops?

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I would only push if there was a reasonable chance to get called. You want to raise an amount that makes calling for him incorrect, even if he stacks you postflop. I would venture to say that this would be a raise to about 1500 (this pretty much assumes that he has an ace and that you pay him off if he hits) but you can do the math if you want something more exact.