two lines.
1. Fold the flop, hope hes holding a better hand (i bet he isnt)
2. 3 bet this flop line? Classic 10nl/20nl line is that if the A or Q flops ppl just instantly fold putting you on it (unless there a real donk) which is why you get a lot of players calling with Kxo and Jxo because they dont put you on that card preflop. However the raise is a lot here. Opp probably doesnt put you on the king then and is just bluffing either with nothing or a weaker king or has a set and knows how to get you pay him. As played his check behind on the turn is a monster and the river is a good fold.

One thing id say is do you need to raise 5bbs in lp with a possibly dominated hand? I know that raising here is good, and i like it but do you need to isolate so much. My reason for this is it inflates the pot a bit more preflop which is fine when you know (holding AQ/AK) you arent dominated but with KQ you may be.
Looking at this hand and the way opp plays if plus reads id either push or fold. Calling is the worst decision but feel free to counter that.