Our range is so capped but we still want to probe our non sdv flush draws. V is prepared to call very often though in my experience. With Ax I will usually check again to protect my weaker pairs and bet river big if V checked back or x/c x/decide.

I also don't know how much air BB should be cbetting in theory which should dictate if I should even bet in the first place.

The best I can come up with ott is betting with a narrow polarized range to give myself some fold equity with my draws while getting value from V's weaker check backs. The drawback is obviously that I cap my turn checking range even harder, but maybe this is fine?
Something wrong with my thought process?
Does anyone know what a solver would do here?

PokerStars - $0.10 NL FAST (6 max) - Holdem - 6 players
Hand converted by PokerTracker 4

BTN: 102.2 BB (VPIP: 15.32, PFR: 11.26, 3Bet Preflop: 7.87, Hands: 229)
Hero (SB): 108.2 BB
BB: 100 BB (VPIP: 19.41, PFR: 16.47, 3Bet Preflop: 7.69, Hands: 178)
UTG: 106.2 BB (VPIP: 16.67, PFR: 10.42, 3Bet Preflop: 0.00, Hands: 49)
MP: 248.2 BB (VPIP: 26.32, PFR: 20.00, 3Bet Preflop: 6.45, Hands: 82)
CO: 100 BB (VPIP: 20.56, PFR: 14.44, 3Bet Preflop: 2.67, Hands: 371)

Hero posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.5 BB) Hero has a lot of BS

fold, fold, fold, fold, Hero raises to 3 BB, BB raises to 9 BB, Hero calls 6 BB

Flop: (18 BB, 2 players) 4 4 A
Hero checks, BB checks

Turn: (18 BB, 2 players) 2

Hero bets 18 BB+ with {KJ...98, KJ...98}
Hero bets 18 BB+ with {AQ...AT}
Hero checks with {rest of Ax, weak pairs,air}

The way this spot functions is based on the fact that ip is capped as ****. It basically means that we are able to probe turn much more than we 'should'. A variety of sizings can work as well. You just have to be aware that his range wont fold to a single bet (unless its massive).

Some people will be very sticky even with hands like TT since you will be perceived to have very little value combos. You should realise that hands like A9s are bascially the nuts ott and thus develop your value range accordingly...

I've seen quite a bit of regs check back hands as strong as AJ or even AQ on this flop lol, I think it messed me up long term and made me too passive on Axx boards. I get the idea though and vs reasonable guys I will go big with at least AT+

Checking flop with AQ/AJ is far from terrible since this flop will for sure be overfolded. Most people in the pool wont though so I wouldnt worry too much. In the odd case where that happens, just take a note and adjust accordingly.

AQ-AT/QJs-98s before KQs/KJs, maybe 55 too
idk if we want to pot all these hands tho

if villain checks back hands as strong as AQ/AJ we cant really do any probes because we dont really have a value hand in our range. it would be easier to probe against someone who bets all his Ax hands otf, but against that player we we would rarely get the chance to probe. if villain checks back a lot of aces that means his betting range is garbage or he will mostly check behind giving us free cards to spike a set or make a flush...

Of all the wide and wacky ranges to attack in small stakes you'll choose probably his strongest possible, because he's "capped" (he's not). Almost certainly no air, top set, Ax and big pairs. Yay!

Allowing your profitability to be determined by how stubborn villain is....in the name of balance. Criminal.

(not necessarily directed at OP)