[Stumpy]

Heh, first thing I saw was the flop for the 1st hand and thought "seems like an OK flop for KK."

I don't like any of our options.

I don't see the point in Check/Shoving. All you'd do is fold worse pairs, weaker flush draws, and bluffs. (And get called by flushes.)
I don't think straights are a big part of his range, except 77, but they're not folding either.

I've gone back and forth, but I think bet 600 - 750 / fold to a raise. He thinks you're aggressive, and he seems passive.
He'll call with weaker flush draws, the straight, and maybe even with a made flush hoping you'll bet the river again.
If the [img]/images/graemlins/spade.gif[/img] hits the river, I'd bet small again.