So first of all, its a 1ct/2ct game, which means throw hand reading out of the window. Calling a 3bet OOP is a donk move, but in those stakes they are not even thinking about that.
So Villains preflop range is probably 22+, A2o+, A2s+, KT+, QT+, JT+, ....
You 3bet your KK, which is correct and villain is making more of an ego call aka "I dont let this guy push me off my hand" move, which happens pretty often.
So in these games it's important to remember that it's very often a way ahead, way behind scenario in which hero only has an overpair and 2 outs.
So it doesn't make sense to bet big on the flop. If he has nothing, he will also fold to a small bet, if he has it, hero will lose a lot.
So 1/3 pot is the best option here and betting 3/4 pot would be a massive mistake.
I hate to say it, but IMO your hand is dead after he check raised you. He either hit a set, two pair with T9. I don't see a lot of x/r
bluffing in microstakes, especially not in 3bet pots.
A call would let me think overpair, TP...but a x/r for this kind of size?
It depends on reads what I would do here. If villain was a fish, I'd call, because there is so much trash in his range and he would also "defend ATo" here.Against a reg, I'd just fold.
A word about betting for "protection": It's bullshit. You bet to fold out better hands, get called by worse hands and bloat the pot when you have the goods.
When you bet big in this scenario, only better hands will call and worse hands will fold...not what you want.