If one of those hollow-core doors is a closet door, hang the mirror on the inside of it. Use hollow-wall anchors for the attachment.
One is a bathroom door, the other is the bedroom door. I may be able to find some hook apparatus to hang over the top of the door and hang the mirror from it, but it will take some thought and some searching.
The only other door is the one to the linen/water heater closet, and it has a 12” X 12” louvered vent in it, head high. If it were lower on the door, I could hang the mirror there, but I think if I covered the vent, I would not be popular with management!