I figured turning the concept into reality shouldn't be too hard.
All I needed was a few main parts:
Surprisingly the ClocKit managed to use every single pin on the Atmega328 it runs on - leaving me without an obvious control pin for the shredder.
I noticed that two I/O pins were being used for the alarm buzzer - which seemed unnecessary. I reconnected one of the buzzer pins to ground - and the buzzer continued to work fine - freeing up the needed pin to connect the TIP120 to.
About 30 seconds worth of code changes - and my alarm clock was working!
I swapped in a larger 5v / 1 amp power supply after seeing a "reboot" when the shredder fired up (probably voltage sag). Seemed to address the issue.
Tip - check the voltage on wallwarts with a multimeter before connecting them to anything sensitive like a microcontroller. A cheap 5v wallwart might put out closer to 8v unloaded.
The source code is up on Github - but the changes are so minimal - you may just want to start off with Sparkfun's code.