Hi Al
I've done an extensive search but have not been able to find a solitary mention of (the father) Edward BARKER in Wellington or other NZ records. [Still need to check early electoral rolls - these are not available online].
I'm familiar with some Australian states' BDM records, but alas, know little about those in South Australia. [There is however information available on RootsChat's Australian board - see Resources section - as to what is contained in SA BDM records].
I do think that if you are going to make some progress, then obtaining the death record for Edward Thomas, would be a good move.
There are very few shipping records that remain, for voyages to and from Australia ... and it's more likely that you would find passenger details recorded at a "port of arrival" (than at a departure point). Newspapers of the day, often carried such passenger lists ... but bear in mind, that in many, passengers names were not actually detailed - especially if they travelled "steerage class".
Lu