KK BB facing a 3bet from a passive player

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KK BB facing a 3bet from a passive player

Villain is super loose / passive preflop, will not raise hands like AQ AJ 99 etc. Basically RFI of TT+ or AK & even AK i am not sure he raises or not, which makes his 3 bet range really tight.

effective stack is around $1,100 SB covers me.
MP opens $20, btn calls $20, SB 3 bets to $120, Hero KK 4 bets to $250.
MP & btn fold, SB calls.

Against this particular villain I believe I can fold KK preflop at this stack depth if SB were to 5 bet. Live poker is pretty much KK+ any time it is 5 bet in fun games with a bunch of recs.

Flop ($540) Kd-7x-3d
SB x, hero cbet $200.

I figured his range here is basically JJ-AA, AA discounted to 3 combos since he didn't 5 bet. initial thought was to go around $150 here on the flop, but didn't want to be too suspicious and wanted to set up an easy shove on the turn. If I bet $150 the pot will be $840 when called as opposed to $940 with $650-$700 left.

What sizing would you use preflop and on the flop? I did take note the Kd is on the flop, so I don't have any diamonds on my hand. I believe I can check here some times if I had a hand like KdKx on Kx7d3d board.

Also if the flop came say 8 high, is it possible to ever get a way from this hand facing a check shove or have I put too much into the pot already?

Thanks in advance for all the feed back.

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