Just because he'll have odds to call does not make your shove incorrect. Checking would have been much worse, if he's on a flush draw.
Think in terms of equity. Your equity in the hand after seeing the turn is about 80%. 80% of every dollar in the pot now, and 80% of every dollar going in on the turn will come back to you, on average. If opponent sticks with this hand, 20% will come back to him. If you bet $1 and he calls, he expects to win 20% of $11.25 (the pot plus the dollar you put in), so his call is +EV for him, but it's even more +EV for you, see? Betting more is even better, so long as he calls.
It's a common misconception that you don't want draws calling. You ALWAYS do*. If you're ahead, you make money off people chasing. You just make more if you charge them more. Giving pot odds is not ideal, but it is +EV.
This is especially clear when seen in contrast to your other line: check and shove a non
river. That will NEVER get called (if he's on the diamond draw). A turn shove will ALWAYS get called and hold up 4/5 times.
*- ok, it's no-limit holdem, so there's always exceptions*. Someone with a strong enough combo draw might actually have more pot equity than you on the flop, so you lose money if there's betting.