You know, I really doubt that he would. One of my best friends knows Cody really well and trained with him for several months. This was maybe fifteen years ago. At any rate, my buddy never knew Cody to wear shoes no matter what. There were some lively scenes when they tried to get into certain restaurants.
But it probably depends on the conditions. Cody's no dummy. I was him on a show where he was in sub-arctic conditions, and it seems like he was wearing pants & more regular clothes. Obviously even the Eskimos don't run around barefoot and in short pants when it's -50 F!
There are some parts of the show that are obviously "big time wrestling" aka manufactured drama, but I think Dave & Cody balance the entertainment requirements with the desire to teach in a pretty effective manner.