Pre-flop I would rather 3bet to build a pot (it's going to be hard to get 225bb in when you hit a set after just calling pre-flop) and I think you should be looking for every bit of fold equity you can get here, there's a very good chance you either take the pot down pre-flop or win it with a cbet. Calling is still +EV here but I think 3betting is better.
On the flop I think you should donk bet 2/3 to 3/4 pot. You can easily get your opponent to commit way more than they should with any Q. Check-raising is a good option too if MP isn't weak-tight. The main point being that you have to bet, you've got bottom set which means you don't have the deck crippled at all, it's also a fairly dry flop so KQ will look like the nuts to your opponent. Bet, build a pot, play for stacks.
The problem with the line you took is that there's no air in your range, you've either got the effective nuts or you've got the nuts. 1c/2c players wont be putting you on a range, but even to them check/call, check-raise, overbet is scary as hell. If you take a line like bet, bet, overbet, then they can make a hero call to catch you bluffing or they can even value raise the river with their TPTK.
You also need to be building a pot with hands like this, it's the kind of hand you want to be getting as many chips as possible in the middle with and you need to be thinking early in the hand about how you're going to achieve that.