In my opinion you're 4 betting to get the pot ripe for getting it in on the flop. If the pot is $80 then its going to be tough on flop. If you 4 bet to $80 then he is going to almost always call and then you can GII on flop.
My thoughts are that ANY 4-bet will be treated as the same. I do feel that basically no $1/$2 player gets away from kings pre-flop against one opponent. I'm sure he noticed my 3-bet the previous pot and so that's why I'd conclude that even over-shoving is probably the right play, because if he has kings, he's not folding whether I put in ~$100, or his whole stack.
Now my argument against shoving is that it does seem like in LLSNL, people do tend to be afraid of bigger bets. I did perceive this player to be a nit, but I let that blind me to the obvious fact that if he has kings, he's not folding. My idea to 4-bet to ~$100 would be to induce the shove from kings, not just get a call. I'd be perplexed if they just call a 4-bet with kings in this game. I guess my fear in the moment was that he would have folded, and I didn't get much value with the previous aces, so I tried to get tricky and extract more out of the 2nd ones.
So yeah, I'd say with my read, the conclusion is to shove. I think flatting is fine in the cases that I do get that 7-high rainbow flop. He's never folding kings after the flop, which was my intent on getting it in. So getting it in pre or post, I just favored post at that time. In future similar spots, I'm always 4-betting. But of course, I don't know how often I will be so sure my opponent has kings dead on when I have the bullets.
