Then have your BIOS automatically Startup your PC at a certain time, say 5:00pm.
Make sure you have auto-login enabled, and throw a quick batch script that starts CGMiner in the startup folder. Bam. Done.
Event 1 - Shut down the PC every Monday at 7:00am.
Event 2 - Shut down the PC every Tuesday at 8:00am.
Etc.
The BIOS alarm wakeup only works if the PC is off. If it's already on, it will just keep going.
This lets you adjust what time you want it to shut off every morning, and will also make sure it's hashing nonstop over the weekends.
OK, that seems like a complete solution. I'll test it out soon and let you know. If it works I'll send you the bounty ( or I can donate to your cause, up to you ).