What is Carrier Shipping and why is it required?
Carrier Shipping or Carrier Calculated Shipping is a module within your Shopify store which facilitates your store's ability to retrieve rates from an external shipping app. This is often confused with traditional carriers such as Australia Post, TNT, Fastway, etc.
You don't need carrier shipping to use Australia Post Shipping.
The Australia Post Shipping app does not require carrier shipping and can be used on all plan types. It doesn't require you to have an Advanced or Plus plan which supports Carrier Calculated Shipping by default.
Addition worked with Shopify to remove this restriction.
The only requirement on a specific plan is that your store is not running in development mode.
You can read further information on the topic of Carrier Service over in the Shopify knowlege base.