5NL FR - KK facing preflop reraise

This is at the same table as my recently posted KK hand, where I had a terrific run of cards over the first few orbits at the table. This is the first orbit, and third time I raise pre.

Opp has folded all his hands so far so no reads at all. He may have been enticed to 3-bet with his entire opening range because of my loose image. So I'd put him on 88+,AQ+ for the reraise.

Question is whether to 4-bet or call. I know that calling 3-bets OOP is generally not advisable but I think that 4-betting probably polarizes his range a lot. He may well only shove over with KK+ and call with QQ, AKs and fold the rest.

So 7 combos shove over and if I believe my ranges I'd have to fold. 8 weaker combos call and all the rest folds, so I am only getting value from a really small part of his range and would in theory have to make a tough fold if he 5-bet/shove (that's theory because I'm never folding this in reality).

If you 4-bet, how much and why?

No-Limit Hold'em, $0.05 BB (9 handed) - Hold'em Manager Converter Tool from FlopTurnRiver.com

saw flop

Hero (MP1) ($5)
MP2 ($5.50)
MP3 ($3.95)
CO ($3.94)
Button ($5.06)
SB ($4.37)
BB ($4.26)
UTG ($8.77)
UTG+1 ($5)

Preflop: Hero is MP1 with K, K
2 folds, Hero bets $0.20, 2 folds, CO raises $0.50, 3 folds

Hero?

edit: I realize that I don't have nearly enough reads to put him on such a tight shoving or calling range, but for the sake of the discussion, let's pretend I know he is a 8/8/2.5 3% 3b nit.