It will not be easy, as award seats for next March were released two+ months ago and seats, especially in the premium cabins! were snapped up at that time. You will now be searching for leftovers.
You can search yourself on the Alaska website for all but Cathay Pacific flights. Just tick the "use miles" box and search one way at a time. Use LAX as the start as you can easily add it to your award booking later; starting your search from SAN adds a layer of complication. Qantas and Delta fly direct to Sydney; Korean Air and Fiji go via Seoul and Fiji, respectively. Delta charges as many miles for a one-way flight as a RT, but if you see something in business class you could call and ask the agent to book that and then he or she can search for a return flight. Qantas also flies direct from LAX to Melbourne and Brisbane and you can easily add the flight from there to Sydney. Spend a couple of days and enjoy the different city.
And watch out for "mixed cabin" flights. These appear in the Business Class column but have a little blue seat icon next to the button. Click on that and you will most likely see that the flight offered is LAX to BNE or MEL in economy, and then BNE or MEL to SYD in Business. Not what you want.
After spending some time on the website and finding nothing you may want to call. The Alaska agents on the partner desk are incredibly helpful, apart from one who told me I was never going to find a flight in business class on Cathay Pacific. (But I did, using the BA website to find available flights in business class and then calling and asking for that specific date.) Before you call, decide in advance if you will want only a direct flight to Australia on Qantas or Delta, or will accept a routing through Fiji or Seoul or Hong Kong. ( People really love Cathay Pacific so consider that).
We are flying to Australia next January on our Alaska miles, outbound in First on Qantas, and returning in Business on Cathay Pacific ( with a two-night stop in Hong Kong). I spent hours and hours working on this, researching possibilities and checking early every morning on the day the seats were released (330 days in advance). But my efforts paid off and we have the seats. I hope you have similar success.