Think I'd prefer to just cbet turn with this hand. As played even though his sizing is large, he can have FDs and worse 2 pair for value so I'd call. Sets are less likely as not only do you block them but would expect a fair amount of raises on this flop. Also people don't always call KQo pre here whereas a lot of his Axs and Ahx will play like this.

Don't agree with your statement that he would 3bet all AQ/AKhh here as they can often get flatted vs UTG.

Im calling But really hating it as we probably bluff catch vs size. But ur High in range, abit underrepped. I hope to se AQ or some random, also small chance he got QQ, aj.

No reason to check turn

Both flop and turn check are totally fine plays. You need to check all your overpairs and AJ type hands OTT and should therefore check a lot if you cbet flop with those at a high frequency, which you probably shouldn't.

In terms of hand strength you have a hand that can bet turn and river for a medium size and it's too thin to XR. That hand class often plays better as bet but I don't think it's that accurate to think like that here given the overall strength of villains range. When villains range gets as strong as it is here you can't bluff many hands and it's easy to get in trouble if you bet too many hands for value.