Tough spot approaching the FT

\$400 live event (series finale), already ITM, 13/162 left. Most of the chips are at the other table; our 280k (average 230k) is second at the table. The only player who has us covered (by 30k) is, of course, the villain in this hand. We just got moved here, so we don't have much of a book on villain, but they seem competent.

With blinds 5k/10k, villain on the button (6-handed) raises to 20k. We defend with Kc5c.

Flop (pot 55k, effective stack 260k): 7d5s2d
We check, and villain checks behind.

Turn: 6s
We check, villain bets 25k, and we call.

River (105k/235k): Kh
Not wanting it to check through, we bet 75k. After about 20 seconds, villain announces all-in. (This puts 415k in the middle, and it's 160k more to us.)

Do you agree with the line taken so far?
What do you do with the shove? If you fold, you have 16BB left and face an uphill climb to the final table; if you call and you're wrong, you're out in 13th for 2.6x buy-in; if you call and you're right, you have 30% of the chips in play and can start eyeing the \$13k up top.

My thought process, and the result, below:

So in order to make a hero call, I need at least two of the following things to be true:
1) There's value I beat;
2) There are missed draws opponent is capable of bluffing with;
3) Opponent can turn one-pair hands into a bluff in this spot.

(2) is clearly true; spades, diamonds, and straights all missed. I haven't got enough of a book on villain to say whether (3) is true here. So it comes down to point (1); are there value hands we can beat here?

K2? Probably wouldn't bet the turn.
76, 65 (turned two pair): might have bet the flop for protection, but also might not.
75 (flopped two pair); might have bet bigger on the turn
72, 62, 52--even from the button, most of these should be getting folded preflop.

AA? Ooh. I could definitely see playing it this way; check the flop for deception, bet the turn as draws develop, then figure I have just a king and might pay off.
AK? Similar idea, but might not go for it on the river. And that's probably the worst value villain can have.

Does this, plus potential missed draws, outweigh the value range (which, let's recall, is only sets, straights (and 43 should be folded pre most of the time), and K7/K6)?

I finally made the call, and villain showed the nut straight to knock me out.

I still don't know if this is a good call or not. What say you all?