Preflop
If it had folded around to you, then QJo is pretty much at the rock bottom of your opening range from this position. For that reason its an easy fold, when someone else are in the pot already. This is referred to as the gap concept. You need a stronger hand to call than raise, because you have no fold equity.
Flop
I can be on board with checking back some percentage of the time, especially against overly aggressive opponents, but I think, you should mainly be betting. There are some gutshots available, and as the main rule in poker, you simply want to be betting and building the pot, when you have a good hand.
You are probably going to be stabbing a ton of the time at a board like this, when you have air, and then you cant always be checking the few hands in your range, that smashed the board hard. You can pick a small size, just like you did on the turn. 1/3 pot with your entire range on paired boards is close to, what solvers suggest. It gives him a tough decision with A high, and you start building the pot against his pocket pairs.
Turn
As played you need to bet a little bit larger, as you already suggested to get more value and be able to get a larger river bet in.
River
When he donk lead for full pot, its a very polarizing line. He either have you beat, or he is bluffing. He is not taking this line with AT or even a hand like Q9. For that reason you have an easy call, but its never a raise.
Results
A bit of a cooler obviously, but this is why, you should not play QJ preflop. You are so often going to have kicker issues, not only when you flop a pair but even when you flop trips, and this is going to be rather expensive.
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