The area of "Rio Grande" has long been important for sugar production. It is in northern, not southern Brazil, in the area that sticks out to the east. The name is more for the large mouth of the river, not the river itself; the city and port of Natal are there. After some arguments with the Dutch, the city grew after c. 1650.

Doerflinger, in the same notes, quotes Captain Tayleur; "This shanty was generally sung aboard those little Baltimore vessels that used to run down to Sao Paulo and back to the United States with coffee- ....."
Speaking of Natal and the great wide harbor called "Rio Grande" by the Portuguese, ".....it was a beautiful place, and the sailors used to love it- and the song was sung by seamen all over the world."
It is an area also noted for coffee, which was grown there quite early.
Lots of "Portagee" girls- and the area still sought by tourists (mostly European and S. Am).

In other words, the area, and chantey, have nothing to do with gold, but with coffee, and perhaps hides, from this area of fine, golden sands and, undoubtedly, some good watering holes for thirsty seamen.

When did Brazil become important in the American coffee trade? This is the likely time for the chantey to have originated. Probably mid-19th c. after the Mexican War.

I doubt that Rio Grande del Norte between Texas and Mexico had anything to do with the chantey.