15$ 9man AJ

How many hands on opponent? Rocks like 9/7 I don't expect 3 bet with AQ only AK and super strong PPs (QQ+). This being said he has tons equity on this flop if we assume that he wold flat a lot of suited AX, Broadway type of hands.
In general vs rocks when you c bet flop and he calls I would play c/fold game on turn as their range is super tight and will crush us long run (if you open wide that is).
On turn when we bet we fold him from weaker hand (some sort of SD or FD) so there is also reason to double barell but again vs 9/7 guy in general we are behind post flop.

77 hands. So you suggest c/f on the turn, and what about the flop?

In general vs rocks I would c bet 100% of my range flop then c/fold turns if I did not have strong equity. In this hand you have some equity but you may have only 3 outs as if he is on a FD you have 12% flop to river or 6% equity turn to river. If he is on a draw you could play for pot control here and take the hand with A high on the river with check back. I don't expect 9/7 on river to bet as bluff but always with completed hands so if he bets the river you have easy fold with A high in my opinion.

note that either a 9 or a king gives me a straight.

I agree, bet once on the flop to fold his low pp. If his fold 2cbet is very low then bet twice. you probably got him to fold a hand like 99 which is a very small part of his whole range on turn. It is really oponnent depended because in this case we see in his range mainly AQ and KQ (because he is tight). Against a wider range I would bet the turn.

I agree with the comments above. I will cbet flop and c/f Turn. This turn is not scary for him, his range preflop is very strong and after he joins us on the flop, I expect him to continue very often on the turn also.