I would say Qantas has the best availability. You don't need to call to see the availability. You need to sign in to your BA account, and search availability LAX-MEL and SFO-MEL. (Click on include partners when it says BA doesn't fly the route.) Once you find some good transpacific options, write them down, and go to aa.com to see availability to get to LAX or SFO.

To fly coach/business/first will cost you 37.5/62.5/72.5 AA miles each way.

With a quick search, I didn't see fantastic availability, but there is a LAX-SYD-MEL on July 4 in economy.