Probably not a ton of rec players at these stakes, so I would lean more towards a thinking player if you feel it is between those two for V. If this is true you then are right to assume his range is a bit wider.  Also, since the only stack that has him covered has already folded, that makes this a great opportunity for V to be aggressive against the shorter stacks.  There are likely many hands in his range that you would dominate, a lot of AK hands that would make it a race, and only two in which you are behind.  

Do you have any idea of how V views you at this point?  Looks like you've played a decent number of hands together so I would think about the image he might have of you at this point.  Your min raise could be viewed as weak or very strong.  Therefore, his 3-bet could be to find out where he stands.  

If he views you as tight then he is counting on this 3-bet to induce a fold any time you don't have QQ+ or AKs and probably expects a shove from those top tier hands which means you have decent fold equity.  Problem is that he only calls the shove with the very top of his range.  I would still say shove and pray he's not in the top of his range in this case as I feel you are still ahead of his overall range.  Seeing a flop is pretty risky with so many Ax and Kx hands in his range.

If he views you as LAG then he probably doesn't have much of a read on your hand strength at this point.  His image of you in this scenario likely widens his range even further so I still think you should be looking to get it in on him.  I would do whatever I could to portray as much weakness as possible while putting more chips in the pot.  That may mean shoving now if you feel he'll view the shove as weaker than a min 4-bet or it may mean to min 4-bet if there is a chance he'll shove lighter against LAG opponents thinking he can still induce a fold.