When I restart Opera, it always prompts on the first try, and it always keeps the '2' and the '999' in the 'count' node. It does stop prompting after that, but seems to eat the keypress (never opens PicPick, even though it always works from every other app I use).

Under root/snap/hint, there's a "count" node of type integer whose value represents the number of times that hint has shown. After the hint is shown twice and it gets set to 2, I don't see the hint anymore when pressing the print screen button on the keyboard (even after restarting Opera). However, you can try setting it to something like 999 (and save your changes) to see if that stops Opera from bugging you for a while. No guarantees though.