Lomax and Lomax, "Billy Barlow" in "Our Singing Country," 1941, pp. 101-102, is the very same song as "Let's Go A-Huntin'," No. 159, pp. 311-312, in "Folk Songs of North America." It is a version of the one Brown collected in 1916. On p. 303 of FSNA, Lomax quotes two verses of "an old English version*" to show comparison but mentions no other American version.
I believe that you are mistaken; if either Lomax collected another version, it is not mentioned in their books.

(Not included in "American Ballads and Folk Songs" nor in "Cowboy Songs and Other Frontier Ballads").

* versions in England and Scotland, mid-18th c. Versions of the Wren ("Hunt the Wren" and "The King") collected in northeastern U. S., mentioned by Lomax in passing (p. 311 of FSNA) but not collected by him.