SB: 9.12 BB
BB: 44.07 BB
UTG: 12.86 BB
UTG+1: 25.28 BB
MP: 23.81 BB
MP+1: 26.15 BB
Hero (MP+2): 26.84 BB
CO: 21.63 BB
BTN: 32.33 BB

9 players post ante of 0.15 BB, SB posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 2.85 BB) Hero has K K

fold, fold, fold, fold, Hero raises to 2 BB, fold, BTN calls 2 BB, fold, BB calls 1 BB

Flop: (7.85 BB, 3 players) 7 9 4
BB checks, Hero bets 2.04 BB, BTN calls 2.04 BB, fold

Turn: (11.93 BB, 2 players) 5
Hero bets 3.1 BB, BTN calls 3.1 BB

River: (18.14 BB, 2 players) 7
Hero checks, BTN bets 8 BB, Hero calls 8 BB

Would you have bet on river? If flush didn't get there, what hands would BTN have to call a bet here?

First off all what do you mean by If flush did not get there? There is no possible flush with the board you posted. Bet size on flop and turn seem way to small.

Sorry, english isn't my first language. It should read "Since flush didn't get there...". Bet sizing is my usual cbet size.

How would you have played the hand ?

I would go for bigger bets on flop and turn too. Imo BTN might be calling you with just a top pair. Even if he folds to a bigger bet on the turn you should get some value from a bigger flop bet.

At what stage of the tournament are you ? Close to the bubble ? Far from money ?

Well in a vacuum, and with the given stack size, that would be my plan :

Let's say villain is calling with 77-33, ATs-A2s, K2s+, Q6s+, J7s+, T8s+, 98s, 87s, 76s, AJo-A2o, K7o+, Q9o+, J9o+ which is 31,83 % of hands.

I would bet 3.5 BB or 4 BB Preflop. So we have more or less a 10 BB pot and 23BB (for simplification) in our stack. On a safe flop like this one I would bet 7 or 8 BB meaning around 75% of the pot with the intention to go all-in on the turn, which is very safe. Because he needs to have specifically two hands to have a straight and only one is in his range. Other hands that beat have remote two pairs or sets. So not much to be afraid of.
Again, it depends on the stage of tournament the type of player and table dynamic.

Is your OR 3.5-4 bb being 26bb deep?
I guess you don't have a r/f range? 3.7bb is around 15% of your stack.

You should've just bet bigger on flop and overbet jammed turn since you don't block clubs and straight draws+2 overs type hands but as played I'm not sure why you're checking river. You've got a slam dunk jam. Villain shouldn't have many 7x hands and if they do they're suited and that means there's 2 combos of it. 999/777(rivered quads)/444 all raise the turn or possibly flop given it's pretty draw heavy and BB is left to act facing a small bet. You're not giving villain the chance to hero with pocket 8's and maybe A9 or 89 which would block our straight combos (yes we should open 86s a non 0%).