AK 3bet OOP pre, missed flop

no real reads or stats, initial raiser raises perhaps a slightly wider-than-average range. PF 3bet is standard for me. i want to take it down and avoid playing AK multiway OOP. i understand BU probably calls/raises with anything that beats me and folds out anything i beat (i'll assume his range is 99+,AK,AQ and that he knows not to setmine with such a high % of effectives in the pot pf) . i just cant justify 3betting and taking the initiative pre then c/f'ing over 3 streets. if my assumption is correct, that he calls flop bets with with TT,JJ,QQ in this hand (or any picture overpair on a flop in any other hand) despite the majority of my range being AA,KK,QQ, should i bite my tongue and c/f AK here, making sure to exploit villian when i do have AA/KK?

if i do have to bet, on the flop or turn?

PokerStars No-Limit Hold'em, $0.25 BB (9 handed) - Poker-Stars Converter Tool from FlopTurnRiver.com

Button ($27.25)
Hero (SB) ($28.70)
BB ($29.90)
UTG ($23.90)
UTG+1 ($38.50)
MP1 ($16.45)
MP2 ($49.85)
MP3 ($25)
CO ($24.65)

Preflop: Hero is SB with K, A
2 folds, MP1 calls $0.25, MP2 bets $1.25, 2 folds, Button calls $1.25, Hero raises to $4.50, 3 folds, Button calls $3.25

Flop: ($10.75) 9, 4, 6 (2 players)
Hero checks, Button checks

Turn: ($10.75) J (2 players)
Hero bets $7.25, Button raises to $14.50, Hero folds

Total pot: $25.25 | Rake: $1.20

I don't play FR. But I CB the flop about 1/2 and then chk/fld the Tu. I think by 3b PF OOP and then check thru the Flop, you're putting AK/AQ in a big part of your range assuming he's paying attention. By the turn you've basically got sh!t... and I don't think you can convincingly rep the flush coming in (and I don't see how he thinks you have AA/KK/QQ after chcking the flop on a 2flush board...) I bet/fold flop, chk/fold Turn, re-eval on River. I just don't like getting jiggy w/ whiffed AK overs OOP that don't even have a draw to help 'em out.

my initial post is a bit convoluted and my logic contradictory. my thinking on the flop was "he's not calling with anything worse and i dont want to build a big pot OOP with ace high" so i check. then on the turn the "hold up i have the initiative in this hand and it feels like a waste of AK to 3bet pre then c/f" idea kicked in which, you're exactly right sarbox, puts my hand pretty close to face up on the table (for any thinking opponent) as i am not checking AA,KK or QQ on that flop.

the turn bet was spewy and caused by the bad reasoning outlined above.

but im still stuck between b/f flop and c/f flop because i think i get called by every hand in his range that beats me and none that dont. what is your reasoning for 1/2 pot on flop sarbox? not that i'm criticising it, just trying to weigh up the merits of both c/f and b/f in this spot.

... and I'm not saying I'm right. I CB 1/2 because my goal here is not to build a pot OOP, but to rep a big PP made hand and force the fold. With 1/2 pot it only needs to work > 1/3 of the time. And I pretty much know I'm not double barreling the turn unimproved unless I've got some compelling read as to why I should. I'm just not big on building big pots that only beat a bluff OOP... I'll save that sh!t for when I'm on the button. But I also want to take advantage of the momentum I built pre- w/ the 3b. Sorry if that's a rambling response... but I'm tryin' to type and play at the same time. Never a good idea for this end of the week....