We're playing live 5/5 NL with a ~$600 stack (shortest stack at table).
V1 (MP) is an older white male who seems to play a little tight, but hard to say early in session. V2 (CO) is a bit splashy laggy, but not crazy.

HERO is QQ in SB.

1 EP limper. V1 in MP raises to $25. Two callers to me including V2.
I 3-bet to $150. V1 and V2 call. Pot is ~$480 and we have ~$450 remaining.
Flop is AK6r and we have first action.

What's our optimal plan for the rest of the hand?

[As played, I checked and it checked through. Turn was a blank (I honestly don't remember, let's say 7h)].

With such small SPR and such a good flop for us this feels like a bet everything we have small or check everything. We've crippled the deck pretty often because AA, AK, KK all leave very little else for villains to have. When we have AQ there is some cards others can have but we are blocking worse Aces and QK too.

I think I like check everything on this flop because there's no scary turn card, our best hands will have opportunities to get it in. QQ and JJ however hate betting because no worse hands call and what are we really bluffing out with one bet? KT/KJ? Probably at least one of these guys has an A is really high.

Game speed I probably bet 1/4 or 1/5 pot which is like a check but probably not as good as a check.

Turn I'll actually split my range between bets and checks and this is a check.

Check and show down and don't put any more money in, the hand for you is over on this flop you just can't win money when you have this part of your range on the AKx flop. Plus it's 3 handed, no more, it doesn't matter that 'you have queens'.

Garbo flop for your hand in a 3! pot multiway. I still really like cbetting and repping AK/AA/KK here though, you hard block AQ/KQ and it's unlikely either V has AK IMO. I'd bet a size to make the most likely Ax fold (AJ/ATs) and just rip it in. Makes Kx fold too. Both Vs ranges are probably mostly semi-good Ax and PPs lower than KK.

I would just c/f esp when a tight older player called a 3bet pre, he's never betting without an ace and if he does bet and doesn't have an ace he's got a king then. A bet or jam otf is just a gamble and by you checking they might also think you're trapping too, so they would likely have you beat if anyone bets, especially in a 3 way 3 bet pot.

Most people don't call our pretty big 3Bet with Ax to fold it on the flop for less than a PSB.

you can check flop and cbet turn small to get looked up with worse pairs trying to 'keep you honest'

Two options:
1) Cbet flop with a small sizing and shut down if called. Given that it’s multi-way and you have less than 1SPR, it seems optimistic but since this board favors our 3 betting range, this is certainly an option to consider

2) Check, fold

I personally like #2 because of the multi-way nature and the fact there is a tight omc.

X multi-way, try to get to showdown. V1 will not bet his JJ/TT.

I like x/f OTF to a sizeable bet. If you have a decent read on how they play post, if it's X'd to the turn, can bet turn and river. This would be very player dependent on who stays + if they're even decent enough to fold a low ace. Readless you're better off just trying to x it down.

RESULT:
I checked turn, hoping to get to free showdown, but willing to call a bet.
V1 bets $175; v2 calls.
I fold. Would have called one bet, but bet/call is too much.

River is a blank. check-check. V1 shows TT and V2 mucks.

I think I botched the hand terribly (and not just saying that because of results). Given PF action and blockers board, I should have acted on strong possibility that both villains had mid-pocket pairs. Should have bet flop small.

You played it perfectly. You just happened to run into the one in 500 situation where you folded the best hand. The other 499 times you lose. Promise.

You're right it does favor our range, so why would we cbet when we dont need a range advantage?

My answer:

1) To potentially fold out the very worst holdings villain might have that beat us (eg KJs)
2) To avoid giving free cards to middle pairs. If I'm facing two middle pairs, that's four outs per street, and it's impossible to know what hits villain.