I had been thinking about what kind of puzzle would be appropriate, seeing as how when I wake up I am barely rational for hours.

If you haven't come up with an approach yet, it seems like the approach of something like Captcha would be appropriate: only difficult enough to engage reasoning, and no more.

You could have a series of very simple problems randomly ordered to do this, along the lines of
- display a series of numbers, and ask which is divisible by 3
if failure, start over, on success, present another like
- show three colored buttons, ask which is the yelllow button
if failure, start over, on success, present another like
- Show several letters, ask what is the first letter of the alphabet
etc.

A configuration option could set how many puzzles needed to be solve before shutting off the alarm.