Bellows won't have to be light tight, but they should look/function similarly (collapsing for storage).
This seems to constrain your choice of antique body ("host") considerably, because when the bellows are collapsed the action camera will protrude backwards into the host. Something like a folding pocket Kodak would probably not be big enough. Since you say you'll be hiking 20 miles with space limitations, I presume you don't want a 40cm wooden box like this one from the 1850s.
However, a shorter version of that design does have some things to recommend it.
The lens only takes up a small part of the front area of the bellows, which is convenient given that (based on the models I've looked at) the lens on your action camera probably takes up less than 25% of the front area.
That, coupled with the fact that your camera is probably rectangular whereas the bellows are square, suggests that extra electronics can be hidden on top of the action camera. That way they don't contribute to the depth to which it protrudes into the host.
You don't have to worry about damaging mirrors because there aren't any.
It's clearly an authentic design for the time period, unlike (say) a Rolleiflex or a Graflex Speed Graphic.
USB is likely limited
Looking at the Campark website, most of their action cameras seem to support Wifi remote control. That is suggestive that they may well support basic PTP. If you have or are willing to spend time acquiring the necessary skills, I would think that it is worth a simple test where you plug the camera into a computer and try to take an exposure with gphoto, ptpcam, or something similar. If that works then my suggestion would be an Arduino project which uses a microswitch at the back of the host as the trigger to send PTP commands to the action camera.
Note: my personal experience with PTP is using Raspberry Pi rather than Arduino, and the ptpcam link above is to my fork which I know compiles for Pi. I'm suggesting Arduino because it should be smaller, and I believe that it has the necessary hardware. However, you should probably have a look on our sister site about Arduino and if necessary ask which models are capable of being USB hosts. A quick search turns up an Arduino PTP library, and a more thorough search might find one or two more.
I'd probably slide the camera into a wood channel.
If your woodworking skills are good enough, that's fine; but it might be easier to get the precision using 3D printing for parts which aren't externally visible.