The workaround is to stab a hole in the pods with a nail.
I figured I could go more extreme...
Step 1: Planning
That's pretty much what I ended up with.
Step 2: Cutting
Using a hodge-podge of sharp things, some of them electrical, I started cutting out the parts of the stapler. I quickly realised that the thickness of the slices of plywood was enough to not need the stabilising leg thing, which is good, because that was going to be an ugly thing.
I carved out hooks using a sharp spinney thing in a Dremel that was probably not intended to carve wood, but, hey-ho, so sue me.
Step 3: Sanding, drilling, gluing and sanding
I then clamped it all together, and drilled through both uprights at once to fit a piece of quarter-inch dowel through them.
I then glued the three base-pieces together, clamping while they dried, and sanded it some more.
Step 4: Pointy thing
To this end, I found a nail the same size as that already being used to poke holes in pods, cut of the head, drilled a smaller hole in the arm of the stapler for a good friction-fit, and then added a little epoxy resin to make sure.