100z 3bet pot co v sb - what should our range look like on river?

Okay so in this hand, I'm not sure of a few things

A) how often people really take this line as a bluff
B1) where I am in my range
B2) what I should be checking back on the turn

My reasoning for checking the turn is we keep all dominated Kx and Ax in the hand, additionally, we get to showdown vs any worse hand that does not bet the river, and for betting the turn if I'd rather have AJ, KQ/KJ with hearts and flush draws for my bluff/semibluff range. I don't really have many value betting hands on the turn so adding AK just makes my range really weird and given the previous points its too strong to start bluffing.

I think I want to be checking Tx mostly on the turn, because betting it always means I have to pay off JJ+ that takes the c/bomb line, and I want to be able to get AQ to showdown without putting any bets in. I think an exception to this is AT, given we block AA (most likely hand to c/bomb??) and dominate KT, QT, JT, T9, T8(if villain is frisky preflop)

I think we should probably call AK *sometimes* - but if thats the case, then this AK must be the worst given we block bluff hands villain would have bet on the flop (AsXx and KhXh)

Against the pool I am inclined to believe this is always a fold from an exploitative point of view because people 1) don't bluff this sizing ever 2) don't bluff this line ever, unless they are a good thinking player (probably more of the pool at 1/2+)

Our range looks like 55,77,88,99 some Tx, some AQ some AK. Agree/disagree?

paired hands total 6+6+6+6 + [ T9s(3) JTs (3) QTs (3) KTs (3?) = 12 ] = 36
unpaired overcards AQ w/ a spade (6) AK (8) - - - AK discounted to account for 4betting sometimes - - - 14

I'm open to any critique on how I arrived at this range. What do you guys think?

PokerStars Zoom No-Limit Hold'em, $1.00 BB (6 handed) - PokerStars Converter Tool from http://flopturnriver.com/

Button ($42.97)
SB ($103.71)
BB ($154.85)
UTG ($105.95)
MP ($166.95)
Hero (CO) ($185.38)

Preflop: Hero is CO with A, K
2 folds, Hero raises to $2.50, 1 fold, SB raises to $9, 1 fold, Hero calls $6.50

Flop: ($19) 2, 10, 3 (2 players)
SB bets $10.50, Hero calls $10.50

Turn: ($40) 6 (2 players)
SB checks, Hero checks

River: ($40) 6 (2 players)
SB bets $25.50