Depart: Jan 15 or 16
Return: Feb 1,2 or 3.

I have 200K Amex and 120K AA. Any suggestions on booking w/ awards? I see AA runs a 37.5K from MEL > LAX on Feb 1 but not anything to get to MEL from LAX in Jan. Any tips?

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.

You might be able to get something via Asia but nothing will come up until a few days prior of at all

There are many threads on FT about trying to get awards USA Australia
You do not state which class you are trying for: economy, business first.
You need to be very flexible with dates and airports. (LAX SFO - MEL SYD BNE)

Airlines that fly USA/Canada to/from Australia (most from LAX) are
- United (LAX SFO – SYD MEL & GUM-CNS) [Star Alliance]
- AA (LAX SYD) from Dec 2015 [OneWorld Alliance]
- Qantas (LAX JFK DFW HNL – SYD BNE MEL) SFO from Dec 2015 [OneWorld Alliance] (AS partner)
- Jetstar (HNL – SYD BNE)
- Delta (VA & AS partner) (LAX – SYD) [Skyteam Alliance]
- Virgin Australia (DL partner) (LAX – SYD BNE)
- Hawaiian, via Hawaii (AA & VA partner) (HNL – SYD BNE)
- Air NZ, via AKL [Star Alliance]
- Fiji Airways, via Fiji (AA & AS partner)
- Air Tahiti Nui (last segments on codeshares) (AA & DL partner)
- Air Canada [Star Alliance]
And others via Asia & Middle East