PartyGaming - $0.05 NL (6 max) - Holdem - 5 players
Hand converted by PokerTracker 4

Hero (BB): 199.6 BB
UTG: 105 BB
CO: 101.8 BB
BTN: 286.4 BB
SB: 100.4 BB

SB posts SB 0.4 BB, Hero posts BB 1 BB

Pre Flop: (pot: 1.4 BB) Hero has 5 K

fold, fold, BTN raises to 2.2 BB, fold, Hero calls 1.2 BB

Flop: (4.8 BB, 2 players) 9 4 A
Hero checks, BTN bets 2.2 BB, Hero raises to 7 BB, BTN calls 4.8 BB

Turn: (18.8 BB, 2 players) K
Hero checks, BTN bets 18 BB, Hero calls 18 BB

River: (54.8 BB, 2 players) Q
Hero checks, BTN bets 259.2 BB, fold

BTN wins 311.4 BB
vs reg of some sort

There's definitely some competing concepts going on that I see. My first instinct was to keep betting the turn. While we pick up some showdown value it's not great and given the flop action I feel like our hand is more of a combo draw than it is a typical OOP mid pair that just wants to see a cheap showdown. But on the other hand the king is just a terrible card for our range since he now has top 2 pair and the top 2 sets in his range and we can't. My solver agrees with your check though, as its turn barrels are A9+ and crappier flush draws, it's checking all it's K hi flush draw hands.

Turn the solver diverges since it's never using a large bet in villains spot, electing to go 1/3 here (makes sense since your range should now in theory be hyper polarized so small bets are the typical size to use against that range). Against the pot size I'm honestly not sure. We are OOP and our outs are not clean. My instinct here is that if this sizing is balanced we might call but in practice I think it's just a fold. We aren't getting direct odds to draw and villain has all the hands in his range that beat us if our outs come through, so that's really bad as well. This is one of those spots where the depth is actually hurting us because even if we improve we are never going to be thrilled calling this bet.