It appears that the two Nuggets (Reno vs. Carson City) shared "Last Chance Joe" as their character-mascot in the early days.

In your first postcard, Last Chance Joe is atop the Nugget in Reno. And you'll find him in neon in the signage above an early version of the Nugget in Carson City -- see this postcard, courtesy of MoGH/ChipGuide:

And even today, the Awful-Awful burger, a feature at the Reno Nugget when it was open, is served today in the coffee shop inside the Carson City Nugget. I had one just last summer.