Darsben1: Interesting idea. The handle holes are about 5" from the edge of the door, which would put the holes through into the bathroom. I could drill through, put in a backing plate and run bolts out. I'll have to give some thought to that.
I noticed the manufacturer only uses screws that extend 3/4" into the wall beyond their bracket. This is a joke. I have 1 3/4 inch of wall. I think my first approach may be to use a slightly larger stainless steel screw that penetrates at least 1 1/4 into the wall. Much further and I could need that backing plate... That will give me more bite and bite into wood that shouldn't have had a screw into it before. (probing with a pick, it seems like the original installer drilled quite a ways into the wall)
On another forum I found the suggestion put some fiberglass resin on several toothpicks and insert them into the hole, pretty much filling the hole with toothpicks and fiberglass resin. Then redial and insert screws. I think I'll combine this with longer, slightly larger screws. If that doesn't work I can go to a backing plate.