“Argle” appears in the late 1500s and means to argue obstinately, to wrangle, possibly a popular perversion of “argue,” or confusion of that word with “haggle.”
“Argle-bargle” is Scottish and first appeared in 1808 in Jamieson’s Etymological Dictionary of the Scottish Tongue. A close variant, “aurgle-bargain,” dates all the way back to 1720. The Scots seem to have a penchant for reduplication. Other examples are: “catter-batter” (to wrangle), “crinkie-winkie” (a contention), “hackum-plackum” (to barter).
As far as we know, “bargy” and “bargle” never existed as independent words. They only appeared as the doubling, or reduplication, of “argy” and “argle.”
An “argle-bargle” has been described as a relatively amicable, if somewhat heated, argument, somewhere between a spirited debate and a fistfight.
From Kidnapped, by Robert Louis Stevenson (1886): “Last night ye haggled and argle-bargled like an apple-wife.” An apple-wife was a seller of apples from a stall, reputed to be just as argumentative and foul-tongued as her male counterparts.
And from Margaret Ogilvy, by J M Barrie (1896): “Ten minutes at the least did she stand at the door argy-bargying with that man.”
The English also enjoy the fun of reduplication. Some examples with conventional rhyming are: super-duper and namby-pamby. Examples of those that modify an internal vowel are: dilly-dally, shilly-shally, wishy-washy, and zig-zag.
Okay. Being namby-pamby, I will shilly-shally around the kitchen, too wishy-washy to decide what to cook. Then I’ll zig-zag to the phone and call a super-duper delivery service.