Jack, that derivation of the Barlow surname from 'barley' is interesting - I suppose it may have been an occupational name originally, possibly a farmer or farmhand's name.
Regarding the Barlow pattern being known as a 'bunny knife' in Australia, here is what I've been able to find.
By way of setting the historical context, rabbits were brought to Australia as food on the First Fleet in 1788, but didn't really take hold on the mainland until the late 1850s. In Tasmania they had been a serious problem in some areas since the 1820s. There were 'successful' attempts in Victoria and South Australia in 1857 and 1859 to release breeding pairs into the wild as game animals on wealthy graziers estates. By 1887 they had spread so fast and so far across the continent, and were causing so much agricultural damage, that the New South Wales government posted a £25,000 reward for anyone able to come up with new ways to control the plague.
https://en.m.wikipedia.org/wiki/Rabbits_in_Australia
So hunting and trapping rabbits for their meat and skins was a pretty common way for Australians to supplement their diets and incomes and for many it would have comprised their primary income. Kids would start off using ferrets and snares, and graduate to utilising traps and firearms. The FN Browning .22 pump action was a favoured rifle for taking rabbits, using .22 Short rounds. Known as a 'gallery gun' in Australia, it's not at all uncommon to see old ones from that era going cheap at gun shows with worn out rifling. Now, anyone who knows their rimfires knows that's the mark of a LOT of use.
Here's a clip of a rabbit drive from 1931.
https://m.youtube.com/watch?v=Ea3u6-w7fno
So consider that all these hunters and trappers and skinners and farmers and kids with ferrets needed to have an inexpensive decent quality knife, that took and held a keen edge for their 'bunny knife'. Sometimes this tool was a sheath knife of a trade type pattern, but I think that would have been more common for the pros. Most people would have had a jack-knife they could carry in their pockets.
Obviously all sorts of patterns were used, but given that Australia was a big enough market, there were specific patterns adapted for the purpose. The one and two bladed clip point barlow with pen blade secondary, and its short bolstered teardrop handled sibling were what epitomised the classic 'bunny knife'.
Looking at some of the old ads, the belly of the clip blade seems to have had a bit more drop and sweep to it than barlows from elsewhere, and the clip is sometimes curved enough that it has a slight trailing point. I couldn't say if these are prevailing differences in the pattern which are peculiar to Australia though, or just manufacturer variations. Some of the clip blade styles would definitely have been optimised for skinning though.
I think the barlow became popular for all the same reasons it did in America - it was a strong, durable, reliable tool for people close to their pioneering roots. The Joseph Rodgers name is still remembered by people of a certain age who wouldn't be able to tell you any other knife brands. Other Barlow bunny knives were Sheffield sourced and stamped with the name of local brands.
How far back does the Barlow 'bunny knife' connection go?
Given that the rabbit plague had become a very serious problem by the latter years of the 19th century, I would hazard a guess that large cutlers and wholesalers would have referred to specific rabbiting pocket knives in their catalogues by the 1890s.
Newpaper advertising copy for cutlers from the earlier part of the 19th century usually refers simply to 'pen knives, pocket knives, and sporting knives.' I suppose 'pocket knives' as a distinct category, refers to what we would now call jack-knives.
These images come from the 1924 W. Jno Baker catalogue. William John Baker was one of the best known cutlers in Sydney who ran a large wholesale and retail import business from 1888 until his death in 1930. They are published in Keith Spencer and Joan Renton-Spencer's 2015 book 'Australian and New Zealand Cutlers and Cutlery 1788-1988'.
(I have cropped and blown-up the relevant Barlow/Bunny knife after each full page catalogue image.)
Spencer believes that the pocket knives with the W. Jno. Baker tang stamp were from the Phoenix Cutlery Works in Sheffield from 1888-1893 and were possibly sourced from Joseph Rodgers after that, given that he also sold a large variety of Rodgers tableware under their own name. (Baker also sold Schrade Safety Press Button pocket knives with his own tang stamp.)
Here's another image from a catalogue of Melbourne company Ward Bros. (I'm unsure of the date, but I'd guess 1940s or 50s.)
So you can see that the Barlow connection in the Australian mind to the classic Bunny knife was definitely in place by the 1920s.
Interestingly, if the secondary blade was a spey, they were often referred to as 'Stock knives', along with the more familiar three or four blade Stockman pattern. I believe this applied likewise to both the long and short bolstered teardrop jack pattern. So in the Ward Bros. catalogue page above, the Bunny knives would traditionally have been both 510 and 513, whereas 509, 515 and 516 would have been 'two bladed stock knives'.
Hopefully more people can add to this information (or correct me if I have been in error). I would love to see some photos of well used old Barlow Bunny knives.
I'd also be interested to hear from any Kiwis if the same terminologies applied over in New Zealand.