There are some amazing craftsmen here who make beautiful things out of fancy materials. I marvel at their talents because I utterly lack them myself. So here's how I would make my own "campaign" concertina. I would actually make all my concertinas this way, and their being suitable for campaigning would just be a byproduct of my lack of proper tools and talent to make anything fancier.
First off, I would make the ends entirely from polyurethane. Rather than machine it (which is possible but a bit tricky and requires tools I don't have) I would build the ends up out of sawn sheets same as with wood, held together with a suitable adhesive. Other than sawing strips to dimension, the only other machine operations would be drilling the button and sound holes. Rather than get fancy with F-holes and scrollwork, I'd just drill a bunch of round holes so the ends would look like Swiss cheese. So that's the face pieces and reed pans. I"d make the buttons by sawing polyurethane rod stock to length. For the bellows, I'd use flex duct of diameter appropriate to the ends, probably held on by hose clamps.
For the actual noise-making parts, I'd buy accordion reeds off-the-shelf, leaving just the action to deal with. The action is a bit more of a challenge. The pads would obviously be polyethylene or polyurethane foam to avoid rot. To rust-proof the action itself (and to avoid all the fiddly work of cutting and bending levers and springs, then mounting them), I'd try hard to devise some pneumatic valve actuation system so the whole system could be made off-the-shelf tubing and fittings.
The pneumatic valve system (I'm typing this as I.m thinking it up) would function similarly to air brakes on a truck. You'd have an air reservoir kept at higher-than-ambient pressure by the normal motion of the bellows during playing. This pressure would keep the valves seated when the buttons were not pressed. Pressing a button would vent air on 1 side of the valve to ambient, causing the higher pressure on the remaining side to push the valve open as long as you had the button depressed. When you release the button, the system returns to equilibrium with high pressure on both sides and the valve closes. Would need a spring to help with this, but that would be a simple, off-the-shelf thing instead of hand-made. Also, before playing, you'd have to give the concertina several pumps to charge up the valve system reservoir .
Note that by typing the above, I just destroyed the patentability of pneumatic concertina valve systems. It's a public domain thing now, so feel free to use it .