Why does the bottom plate on my mac mini rub against the fan?

I've got a late 2012 unibody mac mini. Recently, the fan started making a rapid clicking noise when running. Initially, cleaning the fan and vents with compressed air seemed to help, but then the noise returned. After some experiments, I discovered that pressure on the bottom plate causes the noise; apparently the bottom plate is rubbing against the fan, creating the noise, and slowing or sometimes stopping the fan. If I turn the mini upside down or raise it off the desk, so there is no pressure on the plate, the fan seems to operate fine.

I've checked that the loop on the fan goes over the standoff screw and have removed and re-tightened the two screws attaching the fan, and that doesn't help.

I've disassembled the mini before, most recently to add a second hard drive, so I suspect that I may have made a small mistake while re-assembling it. Any ideas what I might have done wrong or what to check?

Also, until I have a better fix, is there any risk in operating the mini upside down to avoid triggering the issue?