On ranges: With stacks this deep, I wouldn't be all too surprised to find a whole bunch of pocket pairs in his range just because he could figure you to have a monster and he has 11:1 in implied odds
if he makes the call and spikes. In general, people love set mining and will look very hard for odds to make the call. Very, very hard.
But that essentially puts him on either a set or an underpair. I've seen people show up with stuff like QTs in spots like this before, but the strong draw part of his range is going to be miniscule in proportion to anything else, so we don't really need to worry about that.
Versus sets, I don't see us walking away from this hand with any money left in our stacks. If we check back the flop, we're sure as hell not folding the turn or the river after inviting him to bluff. Versus underpairs, we're going to have a hard time getting the money in. Offhand, I'd guess that he's willing to at least call a reasonably sized bet on the flop with QQ or KK because, well, people do that: Peel one off and hope you don't bet the turn or the river. I also feel that if you check back the flop, you've removed any hope of his that your 4-bet was a bluff, because there's no way (is there?) that you'd check back an ace-high flop if you 4-bet light.
My default line would be to bet (and call, if it comes to that) the flop and check back the turn. Also, betting this flop and getting him to spaz out and shove 77 (or some other random hand that he decided he had odds to call with) thinking that you could fold QQ is awesome. Doesn't happen often but often enough to make betting a very attractive alternative to just checking back.
Also, I don't remember if I ever wrote this down somewhere on the web, but I had a thought a few months back where I looked at "fold equity" and decided that the concept, as used, was flawed in situations like this. It's wrong to think of our "fold equity" versus pocket eights in this situation as 2 outs worth of equity in an $18 pot. It isn't. It's 2 outs worth of equity of our entire stack, because if we check back the flop and he spikes an eight, he's stacking us. That brings up the "fold equity" of betting and getting him to fold 88 up from $.81 to $4.30 or so. Not enough to offset the other benefits of checking back (inducing bluffs, chiefly) but certainly adds to the benefit of betting here.
As a sidenote, I don't think he has kings. I don't think many would flat kings out of position, even 200 deep. If this is the kind of player who's afraid to get money in preflop, his 10% 3-bet is a weird way of expressing it. Queens maybe. Tens and jacks are probably the most heavily weighted portion of his range, which is unfortunate since a jack flopped, but we can't do much about that at this point, simply because there are more tens than jacks, and queens and kings are at least likely enough to offset us being behind aces and nines.