Fold seems fine, but how does checking turn save you money vs a set? 

I can’t promise you how any particular opponent will play, but you should realize that your opponent has more incentive to call bets with second-best hands than he has to bet them. Bluff-catching is rarely a significantly +EV proposition. That doesn’t mean it’s generally incorrect, but it’s more of a “lesser of two evils” thing than a spot you should actively be trying to create. What you’re doing here is really the opposite of pot control – you’re giving your opponent the option to get to the river for free when he wants to and also turn your (valuable) hand into a 0 EV bluff-catcher by betting a polarized range. You may be right that your bluff-catching will be +EV because you could be ahead of some portion of V’s value range, but in those cases you almost always do better by betting yourself than by bluff-catching. Save the bluff-catching for hands that aren’t strong enough to value bet.

joelshitshow said
With 14 bigs, what kind of flop are you hoping for when you call pre? I realize it’s 6.5:1 to flat, but how did you feel about your hand with TPGK? Would you have check-shoved the flop or turn if someone had bet?

That’s a good question to ask, but there’s more to “hope for”, as you say, than just the flop itself. It’s a combination of cards and opponent actions that create good situations for you. I’d say what happened here (prior to the river) is basically ideal: Ace on the flop, and everyone checks. You may be thinking that’s a lot to ask for you, and you’re right, but there are other (less) +EV spots, and you can afford to be picky when you’re getting such good odds to see the flop.

I think there’s a subtext in your question that Hero’s stack size means he has to commit to playing for all of his chips if he puts in any post-flop. FWIW, I don’t think that’s right, and I certainly wouldn’t do it by check-shoving: that just causes worse hands to fold and better hands to call.