Plenty of ventilation is the key along with shade. My coops aren't insulated but there is plenty of air flow. The roosts are over a poop pit which I can clean out from outside the coop. All of my coops are different but I can clean them out easily.
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My set up is similar to yours (coop footprint 4x8) insulating alone helped (1½" Styrofoam covered with a plywood veneer).I just hope they don’t think it tastes good
My Mom's people are from Cherokee, OK!! Good luck with your insulation project - please keep us posted on what you did and the result. I forgot to mention my coop is always 10 degrees cooler, but that's not saying much when it's 95.And the humidity there is unreal lol I’m in Oklahoma we don’t have near the humidity you have yay I will definitely get the 2” for roof thank you