*Also another researcher has informed me that they believe they have found an official record for Elias that has been misregistered as Elisa but despite efforts they have not been able to get it corrected.*

NSW BDM will and do correct entries but you need a whole heap of evidence to do so.
Sometimes it is just a transcription error and his birth certificate might have Elias on it even though BDM have Eliza.
IF his birth cert also has Eliza then you will need his marriage and death certs PLUS the death certs of his parents (which hopefully name their children and their ages)

Kellie