That's high season to get down to MEL. It's the Australian Open, and holidays, and vacation season. Economy is usually packed on every airline that operates that route.

If you want a miles ticket during that time, you are probably going to have two options: 1) wait until nearly the day of departure and hope that something opens up or 2) book a standard award for a high amount of miles. You won't find availability now, and I doubt you would have found it at T-335.

You can look to get creative and route into Asia, then use cash to purchase a LCC ticket into MEL. Your cash ticket will be relatively expensive as well, albeit much less so than a cash ticket from LAX - there is no more expensive time to visit Melbourne than during the Aussie Open. I travel for work there every year during the tennis, and our expenses reflect that.