Do you have any more information about his double-barreling frequencies, his aggression in 3bet pots, etc?Without reads, I'm probably leaning to just calling the turn... the Villain is giving us pretty nice odds to hit our draw, especially since you'll be left with over 1000 chips even if your draw misses.  I'd love to hear the opinion of a higher-stakes player on this though... but it just seems a bit unnecessary to shove over the top this deep-stacked if he's giving you better than 2:1 to call. What sticks out to me more is that he min 3bet you three times in a row (presumably a few times before that as well?), and that you are still 2.5x pre-flop raising... it seems min-raising or limping at this point with a hand like K7o (which doesn't play too well postflop) might be a better idea here. 

I think As instead of Ks is a clear call in this situation vs the opponent as described and the only potential difference with Ks is that you might fold off some Ax hands with a turn shove that will beat you if unimproved on the river.But that difference isn't guaranteed (he may call some Ax) and it looks strongly like it could be a small factor here too because he's min 3bet you a few times in a row indicating that his range is much wider than the typical "PP, Ax and strong broadways like KQ" that often are accurate for infrequent min 3bettors.I would call, it's a small bet, you have a very strong draw and even if you miss your draw you still have showdown value with K hi.  And I don't see a lot of potential advantages to shoving here, especially with you being in position.