A lot of interesting responses/conversation.
I think a good next step would be to ask the guys behind the counter if they have the large backstrap, and if they could install it for me to try again. I do have pretty large hands, not bear paws but pretty big, and I definitely found the PPQ awkward to grip.
Edited: I just re-read and saw the "good" group with the Canik was 6", plus fliers, at 10 yards. I'm shaking my head and not sure how to address that. My PPQ will easily do half of that at 25 yards, and it isn't hard to hit a steel IPSC silhouette with it from 50 yards.
To be honest, I'm probably being generous with the group size (most likely less than 6 inches). I had also shot about 100 rounds through my PPS at varying distances before even touching the PPQ or the Canik, as range rules dictate that in order to rent a gun, you must bring at least one of your own, and as I like to practice regularly with the little PPS. Are these just excuses? Probably.
I ordered an upgraded version of that Canik, a TP9SF Elite Combat. It's a little more gun for not much more in terms of price, but the trigger on it is even better than the plain Elite model.
It's due in tomorrow, so one day this week I'll send some rounds through it and post my best group.
I still do want to try the PPQ with the large backstrap though, in case I made a horrible, aweful mistake.