Villain in BB is a 16/10 reg, 4% 3bet, fold BB 82%, over 322 hands. UTG is a 63/42 aggrodonk and SB is a 75/21 drooler, each over about 175 hands. My image is TAG-ish, though I have been opening up to play back at the aggrodonk UTG.
Merge - $0.10 NL (6 max) - Holdem - 5 players
Hand converted by PokerTracker 3
SB: $5.21
BB: $27.35
UTG: $22.19
Hero (CO): $18.46
BTN: $18.05
SB posts SB $0.05,
BB posts BB $0.10
Pre Flop: ($0.15) Hero has 2

2
UTG raises to $0.20,
Hero calls $0.20,
fold, SB calls $0.15,
BB raises to $1.50, UTG calls $1.30,
Hero calls $1.30, SB calls $1.30
Flop: ($6.00, 4 players) 6

Q

Q
SB checks,
BB checks, UTG checks,
Hero checks
Turn: ($6.00, 4 players) 2
SB checks,
BB bets $3.00,
fold,
Hero raises to ???
Preflop, I flatted UTG with 22 since we are 180bb deep, he made it cheap and I have great implied odds with this villain if I hit a set. When the BB raised, I was pretty sure he was doing this purely for value - his 3bet% is low across the board and he is smart enough to know he UTG and SB are probably calling. When UTG called, I decided to flat purely to set-mine because I have position and I'll be deep against both BB and UTG. Also, I was 99% sure that the SB would flat - based upon watching him play, there was no way he would 4bet with anything less than KK+ and he would have already 3bet those. It goes without saying that he won't fold, so I'm getting 3.6:1 express odds on my call.
On the flop, I was pretty suspcious of the BB's check, as this was an invitation for aggrodonk UTG to take a shot at the pot, so the BB could easily have AQ or KQ here, as well as AA or KK.
When I hit a boat on the turn, I'm clearly raising BB's bet. The only hand I am really afraid of is 66 and that is not in BB's range (and UTG, who could have 66, folded first). However, the board is very dry and the BB seems good enough to be suspicious of a big raise here and could potentially get away from AA or KK. How much is the best raise here? If I pop it to $9, that's effectively a shove, since the pot will be $21 and the effective stack $8. A min-raise to $6 might look a little weaker, as that leaves me with $11 behind and he might think I am leaving room to get away if he shoves over me (obv I'm fist pump calling a shove).
Or am I overthinking this and should just shove into him on the turn? I'm assuming that flatting is bad, since the pot would then be $12 and my stack will be $13 OTR (and I'll be hating life if an A or K hits and he shoves)?