I don't know if you won this hand or not, your logic is sound, but your decision to bet turn without shoving makes me wanna ask, when you pick up the nuts, do you want the 50%?
ON the Flop set yourself for a shove OTT, bet more, he can have a lot of Kx in his hand, shoving lessens variance and if shove(bluff) turn you can get a lot of callers like Kx, and maybe get a smaller set to fold, cause after all we could have Kings