KK "Theory" Question

[Kos13]

This hand just happened...

Ultimate Bet - No Limit Hold'em Cash Game - $0.50/$1 Blinds - 9 Players

UTG+1: $157.00
CO: $93.80
Hero (BTN): $99.50

Preflop: Hero is dealt Q[img]/images/graemlins/diamond.gif[/img] Q[img]/images/graemlins/club.gif[/img] (9 Players)
UTG folds, <font color="red">UTG+1 raises to $3.50</font>, 4 folds, <font color="red">CO raises to $12.00</font>, Hero folds

UTG+1 was your typical 26/7/1 donkey. CO was running ~14/2.5/1.25/325 overall and was playing much nittier than that tonight, so this seemed like a standard fold with QQ. My question is: if I had KK here instead, what should I do? Should I 4bet to $35 and fold to a CO shove? Does KK play a lot differently in this specific situation with the original raiser still left to act (ex. should I be 4betting more often because I don't want UTG+1 to cold call for a 3-way pot)?