Whether or not to bet the turn is really close, and with only 25 hands I think it's a toss-up.
That said, I think bet/shoving the turn may be the way to go here. My reasoning is this:
Any hand that beats us is going to stack us on most rivers anyway (if we're not willing to call a river shove, we should probably fold to the turn raise). If the river is the deuce of clubs, are we check/folding? No. We're going to snapcall his river bet. Or I would, anyway. This reasoning goes for most cards on the river. So in that sense, shoving the turn won't cost us much extra the times that he has a ten, because he'd have stacked us the vast majority of times on the river anyway.
But we do have some fold equity, or at the very least we can charge him for draws. And we don't have to wonder what the right move is when the river completes a draw. The fact that you're playing a little deep makes things trickier, but I still think it works out.
The fact that you're deep, incidentally, might be an argument for not c-betting this flop.