This. Apple doesn't seem you want you to get at the DMG, but luckily it's pretty easy to rip it out of the installer app.
I think your problem is that your Mini came with ML to begin with. The ML download on the App Store does work on the Mini; you just haven't actually purchased ML with your Apple ID because it was originally installed on your Mini. Thus, you're left unable to download it. Your only options become: 1) buy it and get your DMG, 2) use Internet Recovery, or 3) find a friend who will give you a copy of the DMG; this would be sort-of legal since you do have the right to reinstall the software. I agree that it's kind of stupid they don't include the full system in the recovery drive, but as a MBA user I completely agree with the decision.