Just by way of brainstorming, here is how "South Australia" sits in my mind. This is all just
speculation, of course.

It has the "feel" of an American song to me, consistent with the
music-culture to which pieces like "the fireman's "Fire Down Below" or
the minstrel-y "Camptown Races" belong. 1840s-50s era, perhaps.

This mix of "heave and haul" that people invariably sing nowadays
smell suspiciously "off." Pick one or the other. I realize that some
source mixed them (in Doerflinger? I forget at the moment) and there
may have been good reason for that, but from my experience it is
usually one of the other. At this point, the explanations just sound to me like rationalizing the from-one-book-and-then-perpetuated revival version. I'll choose "heave," though the song could work equally well for hauling.

I suspect the reference to South Australia was added to a pre-existing chantey framework.   For me personally, then, the geographic reference is a red herring if one wants to discover related songs. And I wouldn't be surprised if something on the same structure, without mentioning that place, turns up. I imagine something like this:

In Alabama I was born
CH: Heave away, heave away
Among the cotton and the corn
CH: We're goin' to Alabama!

Chorus:
Heave away you rolling king
Heave away, Heave away
Heave away you rolling king
We're goin' to Alabama!

I quite like it. I think I'll try singing that some time!

My speculation for "rolling king" is that this work song applied to a rolling operation-- either to rolling cotton bales (e,g, down to port) or to the "log rolling" of which we've heard.

Well, it can't be worse than Hugill's guess that "Ruler King" was a corruption of "Zulu King" in South Africa!