So, I was playing a HORSE Tournament that had about 40 runners, and we're down to two tables (level 13, IIRC). The situation is that I'm the shortest stack at the table where I was recently moved, thanks to immediately getting flop check-raised in holdem by someone who was behind, but rivered his three-outer anyway.
We're 5-handed, and I'm sitting on 14300T, and the antes are 500, bring-in 800, with bets 2500/5000. Bring in is on my left with a 4 (forgot the suit), and I'm holding;
(JJ) K
one fold (can't recall his card), villain for this hand, who has close to 40000K, limps with a 7 and I complete. Villain calls.
Hero: (JJ) K 9
Villain: (XX) 7 5
Hero bets 2500, villain calls
Hero: (JJ) K 9 K
Villain: (XX) 7 5 4
Hero bets 5000, Villain ponders longer than normal before calling.
Hero: (JJ) K 9 K 7
Villain: (XX) 7 5 4 2
Hero ponders and stares at villain's board, wondering why villain put in a call and ultimately decided he's on a straight draw, possibly picking up a wheel draw, too, and he's not folding if I fire my last chips (3800T) and check.
Hero: (JJ) K 9 K 7 (2)
Villain: (XX) 7 5 4 2 (X)

Hero checks, villain checks back

Should I be firing on 6th when villain is never folding, but doing so puts me all-in, or close to it? Cash game, I'm 100% charging for the draw, but I felt tournament play merited a different line. Any and all feedback appreciated.

With less than a full bet remaining just stick it in, you should be ahead and there don't seem to be any good icm reasons for checking

Against a reasonable range villain should have less than 10 outs (flush draw / pair that picked up a straight draw for instance)

He will sometimes call with 2p or thinking his draw to 2p is live

besides any decent opponent would have bet on the end. so you pay but dont get payed off.
did he really think you would check 3 kings with your last money, i guess so.