I think the math is wrong. 5.7 bb's plus 5.6 bb's is 11.3 bb's not 12.3 bb's.
But even at 11.3 bb's I would just shove the turn.
Your 7 bb turn bet is about 40% of the remaining effective stack - which to me is pot committing. Even betting 6 bb's would be 33% effective stack which is still pot committing...
I mean I guess you lose anyway, but at least you don't get called by hands like 8
7
unless Villain makes a mistake.
The other alternative is to bet about 2 bb's on the flop. This puts the pot at 9.7 bb's so you can make a 5 bb bet on the turn and fold if you read it correctly.
But I prefer the 2.8 bb bet on the flop because it leads to a shove on the turn. Not a lot of people 3 barrel bluff and mostly people don't call 3 bets (the final being all in) with just a pair.
I wouldn't check the turn in general because the board seems somewhat draw heavy with a lot of gutters out there. But on occasion I do check here to induce on the river where I have to call because AA has been under-repped.