The phrase “putting on the dog” means: Making pretensions of grandeur; assuming airs.
This was American college slang of the 1860’s. Whether or not it originated at Yale, it was so credited by Lyman H. Bagg who, in his Four Years at Yale (1871), wrote: “Dog, style, splurge. To put on dog is to make a flashy display, to cut a swell”, and the latter expression in the definition could be defined, “to appear important.”
The source of college slang even of today can be little more than guesswork, and to go back eighty-five years is necessarily conjectural.
But it was then that the Blenheim and the King Charles spaniels were at the height of aristocratic popularity. Nothing could appear snootier, more high-toned than those dogs.
Perhaps we owe this doggy phrase to them.