How do I safely install a mirror on an Ikea Besta door?
So, I went with the mastic.

Bad news - between not boing thick enough, and the fact it's pressboard, I didn't feel confident using any kind of screws to hold the door. I went with a pure mastic solutions.

Good news - this in fact a door, so you can take it off. This lets you keep it horizontal, and put weight on it to help the glue hold.

So, I took the door off the hinges and laid it flat. I marked off the perimeter of where I wanted the mirror with blue tape (Blue Painters tape is one of my favorite tools). I thought the surface was a bit plasticky still, so I scuffed it up some with some sandpaper.  I put the mastic on the mirror, trying to keep the inch from the edge. By the time I was done with my pattern, I was almost done with the mastic tube, so I just emptied it. I put some 2x4s and other heavy things on top to help the glue hold (another advantage of having a mirror it on a door). I let it sit for the recommended 48 hr set period and put the mirror on.

The only downside I see so far, is the weight on the door. It seems stable, but the wardrobe didn't seem 100% level. What was OK before, what closed fine before, now stays open a bit with the extra weight. My next subproject is re-leveling the wardrobe.