Whenever I am 200BB+ deep playing 2/5nl I absolutely positively hate playing for stacks when my hand is face up as AK/AQ and the board is something like A
4
4
or in your case Q
2
2
flat flop, reevaluate turn, and try to get to showdown. Also, try not to display any angst or weakness with your call down as you don't want to encourage villain to bluff for stacks unless you have some super soul read and are trying to induce to play for stakes...
Basically, in spots like this when deep vs villains with an insanely wide range on bs paired board where our hand is face up as TPGK or an overpair...
you don't want to play for 200BB stacks and you want to try to get to showdown. So flat the flop raise, reevaluate turn, and try to radiate strength.