It turns out that they do now! You can update your (relatively new) GoPro with a beta firmware build that allows, among other things, to start/stop recording when the external USB power is applied/removed.
I wired up power from my nav light circuit to feed the camera and tested the new feature (enabled when the camera reads a custom QR code) ... works great, and the battery stays nicely charged, too.
https://gopro.github.io/labs/control/usb/
Turns out this 12V to 5V dc-dc converter (to supply 5V USB power to the camera) mounts perfectly onto the two tooling holes across the spar at the end of my -7 wing. No new holes needed.