Recognize the problem? Add it to crontab instead! Just edit your crontab (crontab -e) and add something like this:
25 3-7 * * * /System/Library/CoreServices/backupd.bundle/Contents/Resources/backupd-helper &
This means that backupd-helper will run between 25 minutes past 03 to 07 (3-7 AM) everyday. backupd-helper is the same thing that is run if you would press "backup now" in the Time Machine preferences.
Oh, in time machine prefs, set it to NOT run (since the point is to run it by crontab). And no, backups will not be taken if you computer is asleep (since cron would be asleep as well..)