Weird restriction keeping me from updating to Mavericks?

The message shown in the screenshot is almost certainly not an Apple official message for two reasons: they never refer to the Mac App Store as the 'Apple App Store' and the icon/text layout isn't correct per any other OS X message I've seen.

The most likely cause of the restriction is the JAMF MDM software installed on your Mac (based on the log from your last question).

Therefore, the software that is preventing Mavericks install can't prevent it if you're not booted into that OS. You should be able to create an install media using DiskMakerX. Then, boot into this install media and install Mavericks that way.