nh

You mention, Villain's range were capped. Why do you think his "capped" range is relevant here?

Do you know the fold equity you need for your bluff - and if you achieve that? I'd say it's quite easy for Villain to defend against your shove, given the odds he need and his perceived range.

Wether the bluff is okay, depends on your overall range you get to this spot with. I would not get there with AK. I don't like the check on the flop with 76s neither.

Overall, a hand like K8, A8 etc. might suit better due to blocker effects.

Like the 3bet, dont like the check on the flop, we should be betting 14-18$
Turn check call its OK but its too often spewy against a good reg, and i dont think that you can rep AK too often..
K8 and T9, T8 are better bluffs

It's popular and good to Cbet entire range on flops like this.

I think both betting small or checking are fine OTF, I kinda prefer betting the bdfd + pair combos and check most of the time the naked pair ones.

Regarding to the river, I get your idea of trying to rep AK given your postflop line but at the same time AK is pretty much the only value hand you are repping. 89s, K9s (if 3b) , QQ are hands that you must be betting on earlier streets so I believe that 67s should remain on your check folding range.

Hands that you can use to take this line depend on how you construct your 3betting range but seems like having a 9 or a K on your hand should do a good job on removing his straights + his better bluffcatchers against your line. I expect you to be betting T8s, T9s OTF so I´d say that KTs no bd or 79s are prob the best holdings of your range to bluffshove the river with.