Great point. We just saw one comment right after your reply to prove your point spot-on.

It's always a challenge to think what might have happened if we played the hand differently. Would a turn jam have folded out his hand? Maybe, but if not, and he calls with a draw, you still lose on the river.

Alternatively, if you checked back turn, would he have folded to a river jam, or would he have doubted you'd check back turn with an ace or better, or that you got to the river with a better Kx, and therefore looked you up?

Easy for me to sit here and second-guess you. I do think a smaller 1/3 pot bet on turn and a river jam had the best chance to get through, but I can't lie and say I know for sure that I'd have found that precise line in game. And I suppose we could debate how credible that line would seem, had you over-limped pre, as I and others suggested.

Mostly what others said (limp or fold pre, fold flop, don't bet turn), but I'd also lean way more towards check behind on the river than anything else. There's a reason people don't turn pairs into bluffs much at 1-2 ... V will call any Ax and probably way more Kx than you'd think because he's played any2 from the blinds and has a "good" pair now.
V will also have way too much trash like J7s/65o that you just win against when you check back.

There's not even many better 6x hands you can get to fold (A6/K6 are both the nuts), dito. if he called flop with a good 2nd/3rd pair like A4/A3.
I doubt V ever raises Ax on turn, in fact I wouldn't be shocked if he plays a bunch of 2 pair this way (esp. 64/63/43 but maybe even A3/A4/A6 too).

Maybe you can get 77 to fold.

Preflop doesn't lose much. There are some flops you hit hard. Then if it comes K74, r for example and it is checked to you, you have a very profitable 1/3 pot cbet. As the preflop raiser, you can have AA/KK/AK/KQ/KJ and then 99-QQ may be ahead of whatever they have. On this particular board, you also have a gutshot you might hit.

However, raising larger if you are going to raise is better, to take down the limps or get it 2 or 3 -way. Then you can bluff certain boards you miss.

When there is a bet and call on the flop, you can't continue, even though you are sometimes ahead. Any 6 and small overpairs are ahead and they should have draws if they are behind.

Fish could easily have an A or K. You can't really represent an A or K, because you should have folded most As and Ks on the flop. Plus your sizing preflop looks like a pot builder with a small pp, suited broadway, etc.