How do I improve this spot? (unable to price out a draw)

SB ($9.71)
BB ($10.32)
UTG ($33.23)
Hero (MP) ($31.95)
CO ($76.11)
Button ($25.35)

Preflop: Hero is MP with K, K
UTG bets $1, Hero raises to $2.50, 4 folds, UTG calls $1.50

Flop: ($5.35) 5, 6, 3 (2 players)
UTG checks, Hero bets $3.57, UTG raises to $11.50, Hero raises to $29.45 (All-In), UTG calls $17.95

Turn: ($64.25) 2 (2 players, 1 all-in)

River: ($64.25) 4 (2 players, 1 all-in)

Total pot: $64.25 | Rake: $2

Results:
UTG had 10, A (flush, Ace high).
Hero mucked K, K (straight, six high).
Outcome: UTG won $62.25

Villain is 23/11 and folds 50% or so to 3bets.

I suspect he is completely unaware of position, hence his UTG range to open is basically the same 11% or so of hands he'll open from anywhere, and if he folds half that range to 3bets he is probably calling 3bets with something like {AK, AQ, KQs, TT+}

His checkraising range at the flop is {QQ+} for 6 combos we crush, 6 we split with mostly and 6 that are crushing us. The stove tells us we have 51% equity against {QQ+} on this flop. Perhaps not a great spot, but if he'll checkraise with JJ too then it's 63% and if he's loose enough to do it against the preflop 3better with TT then we have 70% equity.

What he actually did was far looser, and more aggro than I'd give him credit for, but it was basically just spewy - he checkraised the nut flush draw. Why? Who knows. I doubt he figured I'd fold what I had and was repping.

I would never put ATs in his 3bet calling range, but obviously I should have done, and obviously I have made a note on him so I will know in future that this is in his range if a similar spot were to come up.

Anyway, it does no good to try and reason out the actions of this kind of player, and perhaps I was the bigger fool since I did not expect, until I analysed this hand, for him to have such good equity against me.

With $20.42 in the pot and $19.23 effective behind, when we ship it in we give him 2:1 to call with his flush draw. More than enough, since he also has the ace, for 12 outs and 43% equity.

I guess there's not a lot I can do about that - effective stacks weren't big enough to price a draw out.

Should I have called the flop rather than shipping it in? I know mathematically that might be slightly superior, but once nearly half my stack is in the pot already, my decision is going to be difficult if he ships a club turn - I'd rather ship the flop even if I'd known his exact cards. Of course, there also is nothing wrong with getting it in with 56% equity.

Incidentally, although it doesn't particularly tie up with his fold-to-3bet stats, if we give him a range based on this hand of {AcXc, QQ+} for checkraising the flop, our equity is 51%.

Is there some way I could have played this better?