A necropsy may or may not show the cause of death, and testing for all sorts of random things could be incredibly expensive. If the op loses more birds then yes, testing and a necropsy would be in order.
Op some states offer free necropsies, make sure to refrigerate the body asap if you want...
Totally intact as in no ruffled messed up feathers? Do you normally close the chicken door? It could be a natural death.
Have you thought about adding avian netting over the top of the run? The stuff is very strong, and not expensive (a 50 x 50 net is about $60 on amazon). If something other...