Any further information on the villain? Assuming he is a reasonable player he will first off almost always 4-bet KK pf and also QQ a decent amount of times to a SB reraise from most decent regular players (depending on your image).
Discrediting the amount of times he has KK here because of the likelihood he 4-bets PF it he will atleast surely never ever cold call that flop with KK. QQ and JJ are the more likely hand he will just call your 3-bet preflop and again with QQ as with KK he never just calls the flop because there are several cards that will hurt him and the pot is already big, also he is just never folding the turn unless it is a A or K making a flop call very bad, there is just no value for him in calling because he will never get action on the turn from a hand that he has beat (with the exception being when he holds KK to a QQ). He is showing that he is committing himself (given all the action) and that he has a hand by calling, making a reasonable player never barrelling the turn. You having the A of diamonds makes the probability of him having a flush draw given he is a reasonable player basically 0. In conclusion there is no value for him in cold-calling flop with any hand you have beat making it so likely he actually has JJ here. In position with top set here it is a very good move to just call to make sure he will get all the moniez from you almost forcing you to go all-in on the turn with kings or aces.
Chances of him having set of 4s or 9s are incredibly low (once again assuming he is a reasonable player) because of the lacking implied odds for him (you having only a square buy-in in your stack) and your 3-bet size being 8.
The key to making a laydown here is actually realizing he is never cold-calling that flop with KK or QQ, i.e the two hands you have beat, he will be repopping there all the time.
That being said I will have to really be sharp and focused and not to any extent being tilted here to check/fold the turn and when I am hitting the nut flush draw there is nothing to do.
I actually didn't notice that the nut flush draw hit when I started writing rendering my reasoning close to useless in this situation but nonetheless I think that in conclusion a very few reasonable players will cold call the flop with QQ or KK.