MP2 had 22% PFR, hence the 3-bet. I don't have reliable stats on the main villain since he was sitting in and out, but it was pretty clear he was very loose/passive.

Poker Stars $50+$5 No Limit Hold'em Tournament - t20/t40 Blinds - 9 players
The Official 2+2 Hand Converter

BTN: t2230 55.75 BBs
SB: t2785 69.62 BBs
Hero (BB): t5875 146.88 BBs
UTG: t2315 57.88 BBs
UTG+1: t2925 73.12 BBs
UTG+2: t6225 155.62 BBs
MP1: t3280 82 BBs
MP2: t4260 106.50 BBs
CO: t3105 77.62 BBs

Pre Flop: (t60) Hero is BB with Q A
4 folds, MP2 raises to t80, CO calls t80, 2 folds, Hero raises to t300, 1 fold, CO calls t220

Flop: (t700) K 3 8 (2 players)
Hero bets t400, CO calls t400

Turn: (t1500) T (2 players)
Hero bets t2405?

Short of the nut flush, I figure shoving puts significant pressure on whatever he has, and still at least 7 outs against anything besides Axhh

your hand is redic face up, villian is gonna play this spot pretty perfectly

if he is loose passive why bet?

no he won't, not even figuratively.

Face up? Hero can have a bunch of hands here. I think its fine since there probably isnt too many Kx in his flat raise/flat 3bet OOP range.

When I'm in this spot I usually repeat a mantra that I can never follow: "Don't bluff the Loose-passive fish, Don't bluff the loose-passive fish"

It's usually not even worth semi-bluffing unless you have ridic equity and you might not...

Should I just not even bet the flop?

Flop is more debatable as you're not risking a huge portion of your stack and this is a scary board for somebody without a draw. On turn you've realized that a passive station has some interest in it and I generally don't think it's wise to commit. I could entertain some arguments for the overbet though as stations can float too i guess...