I would go with Polycarb. The PVC will be soft in high heat temperature condition. I have Polycarb patio cover for 15 years and still in very good shape.
Do you have a link to the hog wire coop setup? I am going to move coop to the new setup next spring. I do have an option to replace with the free 4'x8' hog wire coop setup, but not sure yet.
Lowes has good deal on white corrugated roofing 75% off going on now. I purchased 6 sheets and plan to upgrade next spring. It rain so much here that make the run so muddy and stink.
I have plywood shipping box coop on the cement blocks for about 6-7 years. It should last for awhile. I spend about $75 for the hinge, roof material, 9 cement blocks, etc.. The run is free chain link fence. You can get most free item from the Craigslist. The trampoline probably should work...