It turns out there was one Kirby mower starter winder and cap from the 1960s that will fit the Southern Cross and is virtually identical. I will post up the part numbers when I have them. The problem is of course that Kirby mowers from that era are also getting rare.
If it's a wind-up [i.e. impulse,
not recoil] starter, it's extremely likely that SC used the same units as found on Kirby engines, including Kirby-powered Victas of that era. Victa used the Kirby part numbers, so they can often be found via that route.
The other thing about these starters is that they were US-designed, so there are Lauson or Tecumseh part numbers for many of their bits.