Nice discussion. Hero has three options, as check-fold seems pretty bad. First of all why did hero call preflop? As the hand is far from great (or even a good one) I don't think hero can call just for the hand value. So to make this call profitable, hero has to win some pots without having the best hand (bluff). Se when we actually hit something, folding is terrible. The options are then donk, check-call and check-raise.
Donking
The problem in donking is that it isolates us against a strong range. And we don't get ANY value from his air. Let's use PokerJuice to figure out how often he actually folds here. He opens from the BTN, so let's assume his range is 60%. On this flop he will call any flush draw, any J+ (J, overpairs, 2 pairs and sets), and any straight draw of 8+ outs (OESD and wraps). He has those 59% of the times. (if his range preflop is 90%, he has those 57% of times, so not a big difference)
So 40% of the times hero wins the pot right away. Donk of 3,5bb has to work around 40% of the times so hero is not making an instant profit here. Against the calling range hero has 53% equity on average. So he doesn't really crush his opponent. And being out of position and having a bad visibility probably means that villain can make better decisions with his range than hero on the turn/river. So donk itself is not immediate profit and playing turn/river is most likely slightly -EV, it seems that donk is not a great option.
Check-calling
When we check, we can assume that villain bets most of his range as the board is quite good for cbetting and we don't have history of hero check-raising a ton (which could make him check some of his hands). Against his whole range hero has average of 64% equity, so it seems good to keep his range wide. The problem is that once hero check-calls, his hand looks like a weakish made hand or a draw. Pretty much exactly what he has. Hero also has 0 nut outs, although J or 8 are quite close to it.
How is hero going to play the turn, if J or 8 comes? Check? Then villain either checks behind with hands that has some equity against hero or barrels and hero gets one bet as value (as villain folds to a raise and doesn't barrel river often). So that 10% of times that J or 8 comes, it looks good for hero (although he can't get 2 streets of value). K is also nice, so let's say that 7 outs help hero, which comes 16% of times.
Diamonds are nice, although what will hero do if he hits his flush? Check-raise is overplaying, check-call opens the hand quite face up. If villain checks, we don't know if he checked Qhigh flush or tries to draw for full house. What ever he is checking, almost 1/4 times the board pairs and hero can't valuebet his flush. Or if the board doesn't pair, can hero valuebet Jhigh flush on the river? So we see that the flush draw is tricky, as it's hard to get value (often just one street) and there are some reversed implied odds (when villain has Q or Khigh flush and takes more passive line).
So around 35% of the times turn helps hero. What about those 65% of turns that are blanks for hero? Check-fold? Or keep calling and hoping? With check-calling the flop, we are hoping for a lot of things. We hope that our outs come, and if they do we hope we are ahead of our opponent. If blank comes we hope that opponent doesn't bet big, but if he does we can call and hope all of our outs are alive. And we hope the opponent doesn't have higher flush draw, 2 pairs or AJ. Lot of hoping.
Check-Raise
If we check and villain bets 3,5bb, we can raise to 9bb and give ourselves 50/50 odds. So if opponent folds more than 50% of the times, we make immediate profit. Imagine yourself here and someone check-raises you on this flop. What hands can you call that raise with? I would think that any set, J+ with a flush draw, OESD+ with a flush draw and any wrap. Villain has one of those around 14% of times. Meaning he folds 86% of the times.
If opponent re-raises, hero can happily fold. If he calls, his range is capped to draws most likely.