Turn sizing with FD and OC vs 44 BB stack

GG Poker - $0.25 NL FAST (6 max) - Holdem - 6 players
Hand converted by PokerTracker 4

BTN: 44.48 BB
SB: 93.56 BB
BB: 186.88 BB (VPIP: 0.00, PFR: 0.00, 3Bet Preflop: 0.00, Hands: 4)
UTG: 146.52 BB (VPIP: 33.33, PFR: 16.67, 3Bet Preflop: 0.00, Hands: 6)
MP: 65.76 BB
Hero (CO): 136.72 BB

SB posts SB 0.4 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.4 BB) Hero has Q:spade: J:spade:

fold, fold, Hero raises to 3 BB, BTN calls 3 BB, fold, fold

Flop: (7.4 BB, 2 players) 2:spade: 4:diamond: T:club:
Hero bets 2.48 BB, BTN calls 2.48 BB

Turn: (12.36 BB, 2 players) 7:spade:

I should want to cbet here and make sure I can call a shove so I should bet bigger. Alternatively, I could check-call, depending on size. But A c-bet profits from FE, and his range is still quite wide I believe (something like 22-88, A2s, A3s, A4s, A5s, 65s, 54s, T9s, JT-AT, and probably some other stuff)

So I think I should make an overbet, so I still get the odds if somehow he has a set ott. But that would have to be 2xpot or above. I think he will shove all sets, some Tx and maybe some 7x, although it is hard to see which 7x should call the flop (and why he should shove it lol), 6s5s. As4s would be bad but I still would have 6 outs.

I am not entirely sure about his range and what he might shove against a cbet. I think he probably raises my turn bet very rarely, as sets and good Tx he will be inclined to raise on the flop. With the sets he probably should not, as the flop is very dry, but does he know that? Hardly any 2pair make any sense ott (maybe Th7h, Td7d or 7h4h, 7c4c).

If I consider a fold after a raise, cbetting small makes more sense, of course. So maybe my line is not bad if I am prepared to actually fold to a turn shove, but I am inclined to think I should not be.

If I get called and miss, what do I do in either case? Just give up and not bluff the half stack guy OTR.

As it turned out, against his specific hand and player type it was probably max EV as played (?).