JJ, AK, maybe QQ (doubtful).

I agree with this.

The dynamic created by this particular spot being UTG, UTG+1 and UTG+3, I feel like it's a lot more difficult to not look SUPER strong no matter what you do.

Another reason for flatting...I think getting 3:1 you could easily justify calling with hands worse than AA that you could have also 3-bet preflop (1010, JJ, QQ, AK, AQs). Therefore I don't think your hand is face-up if you flat. If anything, I think villain's hand is more face-up after he cold 4-bets from middle position. I think I'm flatting with the intention of cc or cr almost any flop.

That or make a super small 5-bet that's impossible for villain to fold to in position and then take him to value town on all non-K, non-Q boards

Should the second KK have folded last night? Seems like it.