Wow! I can't believe what looked like such a straight-foward play has generated so much attention. I read and didn't comment because I didn't think there was much to say other than nice job.

I definitely think betting is the way to go on this turn because of
- the additional outs with the overcard
- makes it less obvious if the flush hits and we might be more likely to gain a bet on the river
- free showdown

With a weaker flush draw I'd likely take the free card though unless I had a pair as well. I'm always betting here with an overcard or a pair as well as the draw though.

Turn check vs Turn bet

Using 15% turn fold equity i would like to see (calculate) which turn decision has the highest yield. With the turn card out and using the previous handrange for villain (which he would also donkbet with) our equity is 29.25%. If we bet turn and he folds his worst cards our equity is slightly less 28.2%.

First lets calculate what happens if we check the turn:

By checking turn we give up 15% fold equity (but maybe he wouldnt improve with those hands anyways) and we induce bluffs more often on the river. A safe assumption would be to say that we wouldnt call the river if we didnt improve.

With his donkbetting hands he will have a worse flush draw 10.65% of the times. Lets assume he would bluff on the river with the bad flushdraw atleast 40% of the times. (since we checked behind on river).

So 0.6x10.65 = 6.39% he checks a bad flush draw

0.0639x5 = +0.3195BB is the amonut we win (on average) reaching river and not improving!

I used 5BB as pot size here instead of 5.25 since the rake should be around 0.25BB.

But if we do improve how much do we win?

Lets say he would bet 50% of the times (with Kx and stronger 7x hands or maybe by pairing river card)

Of those times lets assume he would bet/call 25% and bet/fold 25%.

And if he checks and we bet. Lets assume he would check/call 37.5% of the times and check/fold 12.5%

1/4 he bet/calls = 7BB win (not counting our own bets)
1/4 he bet/folds = 6BB win
1.5/4 he check/calls = 6BB win (not counting our own bet)
0.5/4 he check/folds = 5BB win
On average we win 6.125BB

Net win/loss for checking turn?

29.25% we improve and win 6.125x0.2925 or +1.7916BB on average

70.75% we dont improve and win 0.3195x0.7075 or +0.226BB on average

So net sum for checking the turn is 1.7916+0.226 = 2.0176BB on average

Now lets calculate what happens if we bet the turn:

By him folding we win 0.15x5 or simply 0.75BB outright on turn

And by him calling we loose 0.85x1 or -0.85BB outright on turn

so on turn we loose -0.10BB by betting

On the river he still has a 10.65% chance of having a flush draw. Lets say he would donkbet bluff 5% of the times when we dont improve so he checks river with bad fd 10.12% of the times.

So on the river we win 0.1012x7 or +0.7084BB if we dont improve

If we improve lets assume he would:

3/4 he check/call = 7BB (not counting our own bets)
1/4 he check/fold = 6BB (not counting our own bets)
On average we win 6.75BB

Net win/loss for betting the turn?

28.2% we improve and win 6.75x0.282 (- 0.1 for our turn action) or +1.8035BB on average

71.8% we dont improve and win 0.7084x0.718 (-0.1 for our turn action) + 0.5086BB on average

So net sum for betting the turn is 1.8035+0.5086 or +2.3121BB on average

Conclusion?

With 15% fold equity and a lot of assumptions betting the turn comes out slightly ahead of checking the turn. Its still close though and dependent on a lot assumptions. Ofcourse a turn check/raise would seriously damage these numbers. And with that in mind maybe the options are pretty equal?