Let’s go ‘full retard’ and say that it is possible for an aircraft to be simultaneously employed in aerial work operations and charter operations and regular public transport operations and private operations.
CASA have ruled that an operation can be classified for more than one purpose at the same time (CASA Ruling 3/2004). While it refers to operations under CAR 206, it confirms that more than one purpose can (or must!) be considered when classifying the operation, and there's nothing there that rules out a private purpose. It certainly means that an operation can fall under more than one classification in CAR2(6).
An operation classified for more than 1 of the purposes in CAR 206 must comply
with the requirements applicable to both classifications. Generally, a person who
complies with the higher level classification (e.g. charter) will also comply with the
lower level classification (e.g. aerial work). Guidance should be sought if there is
any inconsistency between 2 (or more) sets of applicable regulatory
requirements.