The SJCAM M20 action camera came with a whole bunch of doodads:

Including a waterproof case, some right-angle connectors, and a pipe clamp:

The stack turns out to be about as flexy as one might imagine, definitely a Bad Thing for a bike-mounted camera, and a somewhat more rugged mount seems in order.
A diagram from the M20 manual shows the parts:

Some camera dimensions:
- 40.2 mm wide + 0.5 mm for the Up/Down buttons
- 21.8 mm thick + 1.0 mm cylindrical front curve + 1.0 mm rear screen
- 50.0 mm tall + 4.0 mm cylindrical top curve + buttons
- 21.7 mm OD × 6.0 mm long lens housing, 1.3 mm down from top center
All the edges have neat chamfers or radius rounding on the order of a few millimeters.
Applying the chord equation to the spans inside the rounding:
- Front radius: 162.5 mm
- Top radius: 42.5 mm
The new batteries survive for a bit over an hour, not quite enough for our usual rides. Rather than conjure a fake battery pack connected to an external 18650 cell with a wire chewed through the case, the least awful way to go may involve a relatively small battery pack (with internal 18650 cells, of course) plugged into the USB port with a right-angle cable and a rigid mount holding both the camera and the pack to the seat frame.
More pondering is in order.