Does the E-3 visa category support the doctrine of dual intent?
The E-3 visa is only for Australian migrants who have no intention of permanent settling in the USA. As such when you go to renew your visa you will need to show you are maintaining ties with Australia (such as property, investments or possessions). You can transfer onto an H-1B visa whilst you are in the USA and this does allow a settlement path / Green Card.
Be aware your spouse will lose the right to work freely if you move to H-1B visa.