I don't hate your call pre-flop, but personally I would 3 bet every time here I mean it is 4 handed..
So played as is I actually agree with check-raising the flop but definitely bigger. Raising 60 cents more to him practically makes him call even with a weaker flush draw. I'd say make it $1.40 at least.
Then your pot should be $3.50 or so and that's when I fire around $2.50 personally, if he has Ax with the ace of clubs even a had like Ace 4 isn't getting the right odds
This puts you in an easier position when the river hits because the pot is big enough to check at $8.50 if he does call, then likely you're not getting called by worse on that river so you check and re-evaluate if he bets.
With a 3bet pot I think this likely would go a little different. You raise to $1 with a strong hand and if he calls in position, you most likely fire flop, turn, and slow down by river.
What'd he end up having? Because i'm nearly almost certain he beats you here but find his range to do so kind of thin.