METAL. Polycarbonate breaks down with time, typically faster than a shingle roof, becomes brittle. Worse, it can create a "greenhouse" effect by trapping reflected heat once it warms the ground.
Wood prices being what they are, a shingle roof is much more expensive, has many more potential errors to be made in construction, takes longer to install correctly, and won't last as long. Most shingles are dark in color, so all the radiant heat you get from a metal roof, you will as likely get from a shingle roof as well. Your metal roof will likely have about 1 screw penetration per square foot, maybe less. Screws hold very well. Shingles? You are driving probably 3 per square foot, maybe more if you place 5 per tap in the first and last row, because a chicken house is so small. Every one of those penetrations is a potential leak.
As to condensation from temperature gradients, shingled wood roofs do it too - but it usually happens between the shingle and the decking, where you don't see it, and won't know its a problem - till you are doing a tear off.
North TX should (unless you are NE TX) be pretty arid, so I doubt its much concern anyways - I would skip the insulation - your birds can take your colds - its much more critical that you get good ventilation. "Raise the roof", hardware cloth your big overhangs all the way around, and allow plenty of venting right under the metal roof. Not only will it ensure your birds are draft free, but it will help with radiant heat from the roof panels themselves, whatever you decide.
Have you looked at open coops designs? They are resource efficient and well suited to hot, arid climates.
