As the action is played out and described, I think check jamming the turn is quite a bad play in this situation. The min raise on the flop tells me one of two things:
1a) opponent has 2 pair +
1b) opponent has a pair + jack
Most likely 1A, occasionally 1b.
with Hero flatting the min-raise on the flop, you can rule out AJ, J9, KK, QQ from hero's range as hero is almost certainly 3-betting those hands. You most likely have 1 pair on the flop and some sort of draw (gutter straight draw, open ended straight draw, BDFD+gutter straight draw, NFD+Gutshot). You can almost rule out the NFD+Gutshot, as you'd probably be tempted to just get the $$ in now.
The Turn brings the 9d, which gives you decent equity to continue on in the hand depending on bet size.
When Villain continues this flop, the story being sold is that villain has a hand again that is 2 pair (most likely top 2 and doesn't know if he should check or bet), a set, or a straight (most likely a straight that was flopped).
Start of turn has $115, after villain's bet turn=$155, after hero shove turn= $402. Villain has to call $207 to win 609, so roughly a 1 in 3 to break even situation in the absolute worst case.
Combine all the following factors: decent odds if behind, great odds if ahead, and the fact the Villain has raise the flop and bet the turn. All of these factors leads me to believe that Hero has very little fold equity against your average 1/2 NL live casino poker player. Your average 1/2 NL player at a casino is not raising the flop, betting the turn and folding to a jam of around 100 bb's in this scenario.
Most players will make you show them their set is no good on the turn. Most players will make you show them that their straight is no good on this turn. The real hand that you are targeting to fold is 2 pair. That hand is a 50/50 as to whether it folds or not, but it's also the least likely of Villain's original range to bet the turn. A set or straight is betting at a higher frequency than 2 pair in Villain's shoes.
Estimated Call/Fold Equity (IMO) with hands that bet turn in Villains shoes:
Straight/Straight Flush (has this 40% of the time): 100% call / 0% fold
Set (has this 40% of the time): 97.5% call / 2.5% fold
Top 2 pair (20% of the time: 70% call / 30% fold
By the ranges and estimates I've constructed (out of 100 hands), Hero is getting called 93 times in this spot (40 times by a straight, 39 times by a set, 14 times by top 2 pair) and 7 folds (never folding a straight, 1 fold by a set, and 6 folds of top 2 pair).
In this scenario, obviously hero needs more Fold Equity from Villain than 7/100. Based on range of both H and V as well as available fold equity, hero Should only be shoving "straights or better" on the board.
You have roughly 26% equity vs the range of Villain in this situation (2 pair, straight, set, occasionally straight flush). Hero needs to call $40 to win $195. You're getting profitable odds to call the turn bet, not to mention an Implied odds should you hit. Calling the turn appears far and away the best option here.