KK Top Set on Flop

If you check flop, it doesn't make any sense to check-raise. Also, your size is bad. Make it a little bit bigger so that you can shove turn easily.
Bet yourself or check/call flop.

You are commited OTT by betting, so just shove the 3.69$ you have left, it is a 130% overbet.
Btw, if you check fold top set on this turn you are folding too much for sure. It's very exploitable.

I would 3bet bigger preflop to make hands like he has to not have such an easy call

when you check this board it's very suspicious - why would you check your bluffs here? So you pretty much telegraph that you have a made hand when you check and then give him action

So I'd bet 25% just like I would with cheap bluffs and try to induce spazzes from him
which is really what we want blocking the Top Pairs that will give us action vs the big sizes. I assume that's what you were thinking so you were on the right track but didn't think about what you represent here much

On the turn his range is stubborn random midpairs, floats and flushes that hit and the Kx you block that might fold . I'd check or bet 25% again

3bet larger before flop, raise larger on flop.