[Xhiggy]

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We'd like to get to showdown as cheaply and safely as possible in this hand, not pump the pot.

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A Kxx flop is very good for your hand given the betting so far. You want to be putting in as much money when you're ahead, and raising the flop not only does that, but also (given the presence of the button likely cold-calling), allows you to cheaply get away when you're behind.

If the button is very likely to fold your flop raise, then you should definitely be capping PF.

I see no problem with taking the initiative going into the turn. You're not looking nor expecting to gain folding equity, you're simply trying to get more money in while ahead. You're very unlikely to be bluffed off of this hand.
I take back my suggestion of betting the river (if you raised the flop and bet the turn). However, a lot does depend on how the SB plays postflop, and surely having the option of value-betting the river or checking-behind is a lot in itself.

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The question is did I play my hand perfectly, not the button.

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The way you played the hand, you get less money from the button when he misses, and lose the same when he hits. You've allowed the button to see a flop and turn for only 1SB each. You can make the button make more mistakes (while still putting in almost the same amount of money by the turn) by raising the flop.