Before I read into the other responses for a spoiler reveal or other points of view - as usual I like to keep my response honest and focused on the OP's original post.
Preflop has me a little mixed - we can certainly 3b KQs on some frequency when we are this deep as we are IP vs an EP range that we block the top portion of very well and our OOP villan is going to miss the flop quite often and allow us to cbet and win a large majority of the time and get better hands like Ax and middling pairs to fold - so taking the aggressive initiative here is not all bad but that being addressed we can certainly flat here as well, avoid potentially facing a 4bet and realize our hands
equity by going to a flop.
The thing that stand out the MOST for me is the cold call behind us after we 3bet and EP open -- this should convey alot of strength as GTO strategy will often elect to 4bet alot of hands rather than cold call 3bets -- that said, alot of players (including myself) are not doing this often enough and are cold calling here rather than 4betting. Our villan could have hands like AKo, AQs - AJs - or middling pairs like 99-JJ since QQ+ will likely be putting in the 4bet. Since we do have some heavy blockers I think villans cold call range is pretty polarized to those middling pairs and maybe AJs or an ambitious ATs since we block the AK and AQ combos.
Anyhow - I've been talking alot about multiway pots lately and how we are supposed to play MUCH tighter and passively than usual in them since we are against multiple ranges and villans simply should not be
bluffing almost ever in multiway pots - that includes ourselves as well.
So - with that noted - this hand shows EXACTLY what I am talking about in these scenarios. I know from multiple simple 3 way Sims that we are checking this flop in the middle almost 100% of the time - its definitely in the high 80%s here. Villans in multiway pots ARE supposed to bet in position closing the action and are supposed to bet roughly 30% pot almost always when action checks to them -- this in mind we just get to check/call our top pair good kicker combo.
Now I know some of you are affraid of letting flush or straight draw combos get there on the turn - in reality this is a 3b pot and villans SHOULD not be showing up with QJs and KJs and hands of that nature facing an UTG open and a 3b in front - therefore, it just isn't too likely of a scenario and the only one who could have those combos would be the EP raiser - and again, we block QJs so if we are up against the flush and straight combos it is likely only a hand like AJs and so we just don't need to care as much. So what if the flush card falls on the turn?? We just get to check/call and pot control - the lesson here is don't fear the reaper. You are going to fade those draws more often than they will come into play.
Anyhow - if we DO check and villan bets in the back seat and UTG calls....we just need to fold. Suuuper awkward spot for us to overcall in the middle and there will be a decent chance when UTG makes the call that we are not in lead and even if we are, its going to be difficult to us to face a second bet in the middle. Don't be affraid to fold the best hand once in a while especially if you are playing a multiway pot. The second scenario is backseat villan bets and UTG folds - its an easy check/call for us. The turn card helps us out, but also brings in AJs for the nuts - so easy check/call spot again - and on a brick river if villan triples off at us, we likely have to call it off as long as villan does not bomb about 70% pot on the river. A triple barrel in what was a multiway pot should be very strong, we can still beat AK, AQ and QTs even tho we block those holdings so heavily. Just find it very hard for us to be good in that spot alot - but we are too high up in our range to fold to alot of appropriately sized bets.
Noooow...for how the hand actually played out -- as I mentioned we were supposed to be checking this flop in the middle almost all the time - so when we cbet and get raised in the back seat like this - you HAVE to hear alarm bells at this point...villan isn't raising us heads up...villan is raising into two opponents while stacks are suuuuper deep and their raise sizing clearly shows they are willing to get it in here. Some villans are goofy and don't know how to play post flop properly in multiway pots - but flush draws should NEVER be doing this here - they have too much equity to risk getting re raised and having to fold their hand -- our villan should ONLY be raising with a polarized range of 2 pair or better and should have zero bluffs here. All of this
tells me villan has QTs -- which is HIGHLY unlikely because it would have to be QT of hearts only -- a set of 66s or set of TTs and given that this was a 3bet pot, a set of TTs is veery likely.
Anyhow, now that the worst possible thing has happened to us - UTG folds and we are simply too strong to fold so we make the call. Turn card does help us but as I said, brings in AJs for the nut straight and doesn't change much for us if we are up against TTs here. So, this is a spot where I HATE the turn lead - its an easy check/call spot for us - this is called pot controlling and gives us a chance to see rivers without being raised.
So when we lead turn - and villan raises you AGAIN - this is a clear-cut snap fold...you could not ask for your villan to be any more straight forward with you here...raising the turn lead is always TTs or AJs and we are virtually dead...drawing to 4 slim outs once. Take that to the bank.
So once the worst has happened already and then we decided to one up that one and make it a horrific situation...we decided we hate our tournament life and that suicide by shoving turn is the best play. My question here is that after the flop raise, and then the turn raise, now you are 3b shoving turn - what hands that you are beating are going to call you off?...what's worse than top 2 that's going to call?...NONE!! It's only going to be a set or the nuts because J9s should not be in a 3b pot.
In conclusion this is what I've been ranting about alot lately - you cannot get out of line multiway, it ends up being a disaster. Playing heads up vs multiway are two different animals and players don't seem to recognize this and end up making alot of mistakes. I deduce the most likely hand our villan shows up with here is TTs - and if they got lucky, then AJs (diamonds) is the 2nd most likely candidate.
All in all you're going to lose alot of chips in this hand, but you certainly don't need to be shoveling in 200bb and going broke here.