Preflop
I would also limp, but I much prefer 3-bet jamming rather than making a small 3-bet. You want him to fold here not call you in position, which he will do a lot, when you go this small.
Flop
By 3-betting small you created a 1,5 SPR pot, which is a very awkward spot. Its to deep to jam the flop with just two overcards, and if you C-bet small, like you did, you will be getting 3:1 on a jam from him. Which is pretty awkward, since you are almost priced in to call a jam, if you think, your overcards are live. Then you can check, but you cant really call a bet out of position, so often you are then just putting in 25% of your chips pre without getting to see more than 3 cards. So I dont really like any option here, and my main suggestion is to think ahead and not create this spot preflop.
Turn
You picked up a flushdraw, and as played I would also jam now. You are almost getting the right
odds, even if he never fold, and maybe there is just a sliver of fold
equity.
Results
Against this particular hand I would also have gone broke, since he clearly dont fold preflop, so the result is just a cooler. I will say though, that after seeing his hand, we might start to wonder, if his minraise pre is a sign of a very strong hand like this, which wants action. And if we think so, there could be an argument for just calling the extra blind, see a cheap flop and go from there.