Thanks for reminding me of Finger, whose book I have on my shelf. He writes that he learned the "Australian product" "Billy Brink" in a bar-room in San Angelo, Texas, seemingly around 1900. He prints eleven stanzas of "Billy Brink" without a tune.

Tex Morton (1916-1983), a pioneer of Australian country music, recorded "Billy Brink, the Shearer" in 1940:

https://www.youtube.com/watch?v=rB3jtpKtPu0

Less two stanzas, Morton's text is nearly the same as Finger's. His catchy tune - possibly original with him - is based on "Villikins." It ends with Jimmie Rogers-style yodelin'. (Morton, a New Zealander, sings with a pretty believable Rogers-type accent even on this song!)

Lloyd's version on "Outback Ballads" (1958) has eight stanzas sung painfully slow to a modal tune reminiscent of the one he uses for "Paddy and the Whale." The hero is now "Bluey" and there are a couple of Australianisms not in Finger or Morton, notably "Strike me stone dead!/ "It'll make me the ringer of Stevenson's shed."
John Greenway's tune and seven stanzas on his LP "Australian Folksongs and Ballads" (1959) are from Lloyd, but paced a little faster.

John Fahey's "Australian Favorite Ballads" prints Lloyd's version with one or two variant lines. He recommends singing it "rousingly."

Canadian Wilf Carter covers Morton's version - with extra yodelin' - on his "Sings Songs of Australia" (1969). (This is one of the few post-Lloyd recordings not to feature Lloyd's tune.)

The Australian jig tune "The Wedding of Lochan McGraw," whose first strain has been associated with "Bluey Brink," is of the same family as "The Laird o' Cockpen."