Perhaps a “panda” type bundle bow made with carbon or fiberglass fishing rod material? That would be a very quick bow to build, and all man-made industrial materials, maybe with that Gorilla tape or duct tape to hold the bundle together.
What about a ballista bow? In TBB V4, Tim Baker shows one, basically a torsion frame the archer holds in the middle and fires as a hand held shooting device.
You could build the spring frame out of some sort of plastic / manmade material, lexan or Plexiglass?, use parachute cord or nylon rope for the springs, plastic washers and tenons locked with plastic pins to twist the rope springs, and use some soft of plastic material for the bow arms.
There is a simple formula for designing catapult spring frames, based on the diameter of the torsion spring. I can post that if it may help you.
Also, maybe use artificial sinew in some way, certainly for the bow string, maybe to even lay courses on the back of a bow, something like a real sinew job.
If you win using my ideas, I get half of the non-prize
Dane