Olny One Mexican War Ballad I am familiar with mentions the Rio Grande; Buena Vista, by Albert Pike.
It has no bearing on the chantey.
First lines-

From the Rio Grande's waters to the icy lakes of Maine,
Let all exult! for we have met the enemy again;
Beneath their stern old mountains we have met them in their pride,
And rolled from Buena Vista back the battle's bloody tide;
etc. etc. and similar-