1/2 Holdem, 9max.
CO opens 12, BU calls, I squeeze TT to 65 from SB, BB folds, CO&BU calls.
CO big whale, agro preflop, calling 3bets with wery wide range.
BU is even bigger whale, tilted vs me. Has 3betting range, so he almost never have bigger pair here, maybe 50%JJ only. I've seen him calling 3bet with A4o, 63s.
Flop 632 rainbow. Pot 197, I Cbet 125, CO folds, BU calls.
Turn K(completes rainbow). Pot 447, BU has 350 behind, I cover.
K is a scary card for BU, do you like to shove here for 350, or go for a smaller sizing, like 150(not sure if villain can understand that smaller sizing looks stronger then shove, but must be easier for him to call 6x, and pocket pairs) and ship any river.

Will he bet if you check to him? If so, and if you're looking to GII on this board, why not let him rep the K?

c/c. I don't see what betting accomplishes, this is a horrible card for him to call with a marginal hand.

Is button the type of whale who is more likely to make an overaggression mistake or a calling station mistake?

In a vacuum (without regard to button's play style at all) I like shoving. This card is good for our range and I'll have hands I want to bluff with here.

I think it's pretty clearly a better card than average for V to hero call a shove on (edit: meaning when he's against a standard-issue LLSNL player, against a balanced player there are no better or worse cards to call shoves on). The K cannot have hit H unless he bet the flop with no pair, and if he did that, he presumably has random other unpaired hands that are going to want to bluff.

In general reacting to a card that is good for our range by going "Oh no, now he definitely can't call, better check" is exactly the wrong approach. Rather than removing value hands from your range, add more bluffs in.

Calling station mistake. Also after calling flop most of his range has a sd value, I presume he might bluff with A4,A5, maybe AJ. The way he played he's not likely to turn A2-A3, or other weak sd into bluff, I rather see him checking back hoping to improve or hoping that I've given up.

We are Cbeting into 2 whales, I don't think is a good idea to have a lot of bluffs here. Only AQ-AJ, that we cbet Flop and have a good card to bluff on turn.

I agree with all this logic and I like betting a lot more if we think V understands this concept. But I think this spot is very undercalled in general and for that reason I prefer a check and try for value on the river.