Don't know if anyone else has run into this problem, but my new (to me) 135 had a loose rear-view mirror, at the base where it attaches to the windshield. Here was my fix:
Remove the cover and the mirror
The windshield has a plate glued to it. The mirror has a spring-loaded base that twists onto this plate. At the 2:43 mark in the video, you can see there are three "nubs" at the bottom of the mirror base. These nubs provide tension when the mirror is twisted onto the windshield plate, making it tight. In my case, they were worn. I suspect the PO either had the windshield replaced or otherwise had a camera or radar detector mod - it seems as though repeated removal and replacement of the mirror could easily wear these nubs down.
Install a shim
At the local hardware store I bought 1/4" brass strips of various thicknesses. The one that worked for me was .032. I cut a piece to approx. 1.5" in length, got it started in between the mirror base and the windshield plate, then drove it in with a dowel/hammer
Super tight now.