We've got somewhere between 3 and 9 outs. If we discount properly, I think we're looking at ~5...which I think is certainly enough to peel getting 13.5 to 1 with only one left to act behind us. But yes, it sucks quite a bit when Button raises.
Firstly, I think giving yourself 5 clean outs here is a big much with this many players in the pot. But let's say five, for argument's sake.
Even if we do have 5 outs, our reverse implied odds are pretty terrible here. I think we dump off more bets than we want to pretty often when we spike a non-diamond Ace or King and still end up bad, or get drawn out on by the river. TJ is a real possibility, as is any number of weirdo straight draws from the BB (Q9s, AQ) or QK from the button, not to mention KJ, AJ, ATs or any other random opaque hand from some other player.
When we throw in the chance of a flopped set and a big diamond draw, calling this flop is tough, and often not worth it unless you have some specific reason for playing. If you had a single diamond, it would be a great spot to raise because the pot is so big, but we don't, so I think it is a fold.
In your case, since you play well, your peel isn't terrible. You might be in real trouble if UTG+2 hadn't led the turn. But he did, and you're good enough to fold here, so it's not horrendous.
Wang