As I showed last night Mankind's ability to move water through the skin in the form of sweat exceeds all animals in quantity per time or per kg. This means that the total evaporation can remove more heat from a man than for an animal. If as you mentioned earlier that man can sweat 13.5 liters per day, this is 7830 big calories per day or per 70 kg. A camel can only sweat 8.8 liters per day per 100 kg. This is 5104 big calories per day per 100 kg.
Mankind is more efficient either per day or per kg.
A furry animal simply cannot have as efficient a sweating mechanism as a hairless one. I will agree that a kg of water evaporated will remove the same amount of heat whether under fur or not. But fur will seriously hinder evaporation. I do not have the data for dogs, but I do have it for camels. Thus a shorn camel with fur 0.5-1 centimeters long would evaporate 50 percent more water than would an animal with unshorn fur.