There's an interesting discussion of the authorship of "Big Rock Candy Mountain" on this page about Marshall Locke:

https://ourgame.mlblogs.com/marshall-locke-poet-playwright-songwriter-outfielder-4d528cc7783f

An excerpt:

Other songs followed, on up until at least 1925, when Locke was 68. But in following the chronology of Locke’s career I have skipped over the work for which he is known in music circles to this day, even though he surely did not write it from “out of his head.” That song is “The Big Rock Candy Mountains,” well known to fans of Burl Ives and Pete Seeger. The song may have been published in the late 1890s by Harry McClintock (a.k.a. “Haywire Mac”), as he claimed, but in 1906 Marshall Locke and Charles Tyner set it down on paper and secured the copyright, publishing it via the Rock Candy Music Company of Indianapolis. (The Harry McClintock rendition of 1928, the song’s first recording, was included in the soundtrack for the 2000 film O Brother, Where Art Thou?)
In truth, “The Big Rock Candy Mountains” was written on the wind. Like baseball, it has no lone genius as its creator. A ballad called “The Dying Hobo” was published in 1895, before either Locke or McClintock set it down, but the song may be traced to English and Scottish folk ballads perhaps a hundred years earlier.

The 1935 Catalog of Copyright Entries shows that Charles Tyner and Marshall P. Locke entered a copyright for "The Big Rock Candy Mountains" on November 30, 1934.

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Also take a look at Nowhere in America: The Big Rock Candy Mountain and Other Comic Utopias, by Hal Rammel (the cheapest copy I could find was over $100, so I declined). An excerpt:

...he remains best known for his hobo songs. Haywire Mac's "The Big Rock Candy Mountain" became so familiar during the early 1900s (and had been published several times in sheet music form) that he had difficulty establishing his claim of authorship in 1928. In fact, the song had already been copyrighted in 1906 by Marshall Locke and Charles Tyner. Fortunately, McClintock had printed the lyrics of his song on postcards in 1905 and, after he asked radio show listeners to send him these cards, he successfully received copyrights to both "Big Rock Candy Mountain" and "Hallelujah! I'm a Bum." Assessing the question of Mac's contributions to hobo lore, Norm Cohen writes:
Mac claimed authorship of three of his most popular recordings: "The Big Rock Candy Mountain," "The Bum Song," and "Hallelujah! I'm a Bum." Authorship of all of these has been disputed; if he didn't compose them in entirety, he certainly rewrote them, and was largely responsible for their great popularity through his records, his radio shows, and his sheet music versions. Considering McClintock's considerable skills as a writer, as demonstrated in his prose works, I see no little reason to doubt that his rode in creating these three songs was considerable—granting that there may have been traditional fragments from which he worked.
Norm Cohen: Traditional Anglo-American Folk Music: An Annotated Discography

Here's that 1928 Haywire Mac recording of "The Big Rock Candy Mountains":

https://www.youtube.com/watch?v=ovKk_kPmAk4