We should raise AK pre, 4-bet vs BB's 3-bet, then play cautiously postflop and call the river shove with top pair.
On Jh-8d-9c, BB has a lot of strong draws and made hands, so our range has a big advantage. With AK high and no real backdoor equity, checking back is the best option. We avoid getting check-raised off our equity and let BB stab their air and weaker pairs on later streets.
The Ks on the turn is a great card for us: we now have top pair with a strong kicker. BB leads 10bb into 66bb, giving us a very attractive price. Calling is the standard play—our hand is strong enough to continue, but not strong enough to stack off yet, and we don't want to fold out all their weaker pairs and draws.
River 6h doesn't change much; BB shoves 62bb into 148bb. With top pair and a strong kicker, we're far ahead of their bluffing range and only behind a few stronger Kx and sets. We should call the shove and take the money.