The cycleway=lane tag should work for a dual-carriageway road where the cycle lane is on the left side of the carriageway, i.e. in the normal travel direction for Australia. However, it causes cycle lanes to be displayed on both sides of the carriageway resulting in display of 4 cycle lanes for a dual-carriageway road.
Examining a dual-carriageway road where the cycle lanes do display correctly (Grant St, Cottesloe), indicates that the cycleway:left=lane tag has been used.
See OpenStreetMap - compare Grant St to nearby Curtin Ave.
According to Tag:cycleway=lane - OpenStreetMap Wiki and Key:cycleway:left - OpenStreetMap Wiki the simpler cycleway=lane tag should work where the cycle lane is on the side of the road corresponding to normal traffic direction.
Potlatch 3 (Windows) editor doesn’t display cycle lanes even in OpenCycleMap style which makes it difficult to see the results of the tags. Adding a cycle lane to a 1-way road in Potlach creates the cycleway=lane tag so manual editing would be required to change to the cycleway:left tag.
Please note that I am talking about a perfectly normal cycle lane on the left side of a dual-carriageway - not contraflow or anything unusual.