Game:

Zoda's Revenge: StarTropics II (1994)

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All credit goes to user skater43 for figuring this out.

https://gamefaqs.gamespot.com/boards/587654-zodas-revenge-star-tropics-ii/77635823

From Chapter 5 of "Zoda's Revenge: StarTropics II" for NES. Full text of the saloon song:

EUREKA! I'VE DONE IT. I'VE HIT PAY DIRT.
RICH IS WHAT I AM AND THE TOWN I WILL ALERT

THERE'S GOLD IN THEM THAR HILLS.
AND IT'S BROUGHT ME MANY THRILLS

NOW IT'S GOLD THAT YOU CAN EARN.
BUT THERE'S A STEP YOU MUST LEARN...
IT'S THE...

CACTUS DANCE, DO THE CACTUS DANCE.
EVERYBODY IN THE DESERT, DO THE CACTUS DANCE

STEP LIGHT AND DON'T YOU STICK.
SEEK A SPECIAL CACTUS AND MAKE IT QUICK

FACE THE CACTUS IF YOU MIGHT.
THEN TAKE 2 STEPS LEFT AND 3 STEPS RIGHT

TURN LEFT AGAIN. OK, THAT'S ALL.
THEN KEEP ON WALKIN' 'TIL YOU HIT THE WALL

CACTUS DANCE, DO THE CACTUS DANCE.
SHAKE UP THE SHACK WITH THE CACTUS DANCE

THE LOST MINE IS YOURS IF WANT THAT TOO,
JUST BLAST THE WALL AND YOU'LL GET THRU

YEAH, BOYEE