On the last day of our trip home, the mirror above the bed gave up and cracked across one end. Today I pulled it off to see about a replacement. Here is what I found. The back wall is 1/8" paneling, covered in wallpaper (doesn't look like it could hold screws, it is so thin). Behind that is white styrofoam, at least 1.25" thick (the screws that hold the headboard on are that long). Then the mirror was held on with adhesive, and rested on two approx 4" sections of 1/8" paneling that are screwed into the above described wall. The glue had almost completely let go, so the mirror was bouncing on the small bit of paneling, and eventually on the screw, so when it let go and came down on the screw, it cracked. Luckily, the back of the mirror is covered in a plastic stick on membrane, so the pieces stayed together.
I have ordered a replacement from the local glass shop, but will no longer have the original etching. Not a problem, as it was not anything special. I will install the replacement with a full width support across the bottom, thicker than the glass (I will be putting in 5mm glass, not the 3mm that Beaver used).