Playing at SugarHouse in Philadelphia. 1/2 blinds, $550 effective stack size

Hero is UTG+2 the button straddle to $4 is on.
SB folds, BB calls, folds to hero who has KQ
We raise to $15
MP Villain who I have tangled with repeatedly calls- mid 30s black women, pretty solid player. Has shown a propensity to call down with hands like TPTK and fold to big raises. Her stack has been up and down a lot. Currently has about $650 behind.
Button (straddled) shoves all in for $32, has been buying in for $100 at a time and getting it in prefop a fair bit.
Folds back to me.
I decide to squeeze here, and got to $95 total. Villain flats. Main pot has $95 after rake. Side pot has $126. I am a little put off by the flat call here. She called a $25 raise in an earlier straddled pot and called me down with AQ on the ace high board (I had AJ suited and raised from the cutoff, she had limped.)
The board comes KJ3 rainbow with one spade. I think about checking here but ultimately decide to bet. I bet $150. She tanks for about 30 seconds, looks at her hand, and then shoves all in for around $300 more.
I tank for awhile I feel like her flatting range preflop as probably something around 88 or better and ATs or better, there might be some weaker suited kings in her range as well but I block that and I am not sure.

What can we do? Do I find a fold here? Should I have check called or check folded the flop?

Preflop, if she always 3b preflop with AK, i also might try and get heads up with the shorty. If she is an ak flatter, it’s probably too loose (tho this might be ressults oriented lol). I think a raise to 70ish gets the job done here.

AP, Flop bet is way too big. If you bet $80 and she shoves, you wouldn’t feel nearly so priced in.

I think your 3-bet OOP pre flop created this situation on the flop; I think its easier for her to flat your bet pre flop because she doesn't have to worry about the button player who is all in . so maybe her range here is fairly wide. Do you think she could be semi bluffing here with hands like AQ or Q10? This is highly dependent on the player type and you should be getting a decent price here... and you have back door draws. I think you can discount the AK combos here as well unless she played AK really weird. so I guess you lose to sets and KJ which isn't that many combos. Knowing what her perception of you here is important, idk what your table image is at this moment. tough spot but I'd lean towards a call

your hand has 60% equity against the range you described if you plug into flopzilla

In regards to her perception of me it is hard to say. The very first hand of the table I 3 barrled an ace high board with Ax in position and she turned over AJ. Since that point I had been involved in a lot of pots and shown a lot of aggression and was running over the table pretty effectively (my stack was close to double the effective stack size in this hand.) She might think I am a little spewy because of the first hand we played though.

In regards to equity, I knew I was ahead of her preflop range. But her jam in response to my bet narrows that range at least a bit I think?

If you are ahead of her preflop range it isn't by much at all, I don't see you pushing an equity advantage with the 4bet. Not to mention the button straddler can have a hand. The 4bet is pretty reckless imo and you size it so that MP is priced in to come along with everything. She's cally with position on you and there's a lot of dead money to play for so this isn't going to be a great situation for us. When you both miss MP will have a pocket or A high a lot and you'll have K high. I think flat is best and if MP backraises then you are let off the hook pretty easily. If MP calls behind she's very likely to play straightforward since BTN can have a hand and there would be a dry sidepot.

The flop bet is also way too big and I don't see a huge point to it. Jx will probably find a fold to this size. You can also check the flop sometimes. What worse hands are you really getting value from? KT? QTs? You block both of those and it's a pretty small % of the range compared to a random pocket, suited connector or whatever. The pot bloating really put you in a bad spot here.

Also go bigger pre, 15 is pretty small, and if BTN has only $32 behind it might have been better to go to $21+ so when BTN shoves it denies another player's ability to backraise over the top.

Yes it should narrow the range, unless she could take hands in the bottom of her range and turn them into bluffs but that seems unlikely so you have to think which hands that you could take out of her range when she shoves and see where your equity is at that point. The good thing is you should be able to remove AA, KK , and AK most of the time

More initial pre raise. I'd make second raise $110. It's more clarifying.

As played, is V capable of this move with AJ or QT or KT or same hand KQ? Has V ever once 3 bet? Have you seen her flat raises with AK, KK, AA type hands?

Agree with tmo1120 the 3-bet is bad.

Why? Your 3-bet OOP pre flop is pointless with someone already shoved all in. Generally speaking at 1/2 all-in, a player that keeps buying in for 50BB is shoving mostly pp or Ax. Which means you're already behind. So why bloat a pot with a speculative hand when your probably behind for the main pot AND you could get out OOP verse a player with 227.5BB behind? Too Fancy! This is probably the very bottom of your range.

As played, your range includes* (I assume): AA(6), JJ(3), AK(12), KQs(3)... anything else? KJs(2), QJs(3)? Are you having anything here that you could or would bluff the flop with? (I don't think you should bluff much, if at all, with the all-in already in)

I agree it's strange that villian didn't raise preflop with AK, AA, KK, etc. So hard to range with this action on this flop. But what hands worse then KQs can you have? I can't think of anything. So,... the flop bet is probably bad. Check call is better.

As played,... you kind of pot committed yourself. Probably you are ahead of all-in player and maybe you can discount AK for villian assuming Ax makes a large part of all-in's range. But this spot sucks.

Crying call and never play like this again.

*I assume you check a KK(3), QQ(6) and TT(6) on this flop, and you don't have 33 or QJs. right?