The Carter collection In the Shadow of Clinch Mountain, which is pretty complete compendium of their original records, doesn't list a "Brown Eyes", closest being "Bonny Blue Eyes" or "I'm Thinking Tonight of My Blue Eyes". (Of course, during their years on Mexican radio, they surely sang a slew of songs which they hadn't recorded and I doubt there are many surviving recordings of those shows.)

The Mainer title "Those Blue Eyes" may hint at a link to "Bonnie Blue Eyes", but if there's some derivation, I don't hear it either in the lyrics or in the tune.