A second-level joke is that in reality "barney" is slang for "argument" or "fight", which British slang got from Australia in the 1920s according to Partridge. Its introduction to Britain pre-dates the invention of the cartoon character by about four decades.

In sporting slang from the century before, it used to mean a race/fight/contest that had been fixed or that was in some way unfair, a swindle.

* https://archive.org/details/slangdictionarye00hottuoft/page/...