BNE-LAX-JFK-HKG-BNE is 22,014 miles! So you can't do it via HKG. But BNE-LAX-JFK-NRT-BNE is only 20,803. Since BNE-LAX is 2-class operation it won't have skybeds until well into next year. So to ensure Skybeds, you need to do BNE-SYD-JFK-NRT-BNE which is 21,583.
Also note that the distances I am quoting here are from Great Circle Mapper. These are not the exact distances that Qantas use. For example GC mapper shows my BNE-SYD-LAX-DFW-PHL-ORD-NRT-BNE routing next week as 21,866 miles, while QF calculate it to be 21,909 miles.
This fare is available from TA's or any OneWold airline directly. It is called OneWorld Circle Pacific and is available in 22,000, 26,000 and 29,000 mile versions, with some special 29K versions that include South Americ. It is similar to the OneWorld Explorer RTW fare. The main difference in rules for purchase from Australia is that it requires ticket issue 7 days prior to travel commencement, while OneWorld Explorer RTW purchased in Australia has no ticket issue deadline. There is a 10% cancellation penalty if cancelled after ticket issue and before travel.