The difference in weight between 1/4” and 3/16” plate is about 2.6 lbs per square foot so building the transfer case skid from 1/4” only would have added another 5 lbs. The transfer case skid plate is pretty small.
Normally I would agree that 1/4” is overkill for a skid plate. I’ll have to get this out in the rocks and see how strong it is, or maybe I should put a jack under it and see if I can lift the truck from the skid. Now there is a great idea.
3/16s is a lot unless you are really dumping a lot of weight, especially with bends. I'll be amazed if you really manage to mangle a bent 3/16 skid.