if you think his range is that tight preflop, then it's a pretty easy fold to the 3bet. Like, even if you hit a decent flop like we did, we have no idea what to do. Shoving is probably the worst thing we can do on the flop here, check/folding doesn't seem very good since we called pre.. we're not getting many better flops than an A high one.
I think you are focusing on all the negative things that can happen, like checking and allowing people to draw. While draws are a threat, we can't just forget about all the other stuff that we play against. I think our best option on the flop is to go for a check/jam here or likely a check/call since his stack is about pot sized, but it's not a good option, it's just the best of a bunch of bad options.
basically next time you are here ask yourself firstly, what is his range? If you can't get a decent estimate then ask yourself, if i call here, am i happy to play a big pot if an ace high flop comes? if the answer is yes, then call, if the answer is I don't know or no, then fold.
the biggest problem with your shove is that flush draws are likely to call anyways, since players are bad and this is a low limit tournament7. Also, you fold out some stuff that might shove or fire a c-bet, like KK or some random pair. and you get snapped by anythign better. Sometimes you might also get called by something weaker, but it's pretty unlikely since we assign villian that tight range above. It's a bit of a worry with the flop getting checked back, but again, ask yourself if the diamond hits the turn, is it that big of a deal? it makes our decisions pretty easy and we get out of the pot cheaply..