There are at least three different tunes called the Bartender's Polka, and they're all good!
First up is one with Polish lyrics, just posted yesterday! I think of this as Chicago style polka.
I remember hearing Walt Solek's "Bartender's Polka" when I was a kid:
The Bartender's Polka that Hank Thompson had a country western hit with in 1958 is popular with the Dutchman style polka bands that I love. Here is that tune, covered by Gary's Ridgeland Dutchmen:
At the beginning of this video, Gary Brueggen says it's the same tune as Fingertanz. I heard that and thought great, we've got that in our new books! — but it seems to be a different "Fingertanz".