This may be a bit of a stretch, but many airlines (e.g., Quantas) offer an unaccompanied baggage service for very reasonable prices. The catch is that Caught@Work would have to find a helpful person who was flying from San Diego or LA to Oz, willing to pick up the shipment and arrange the unaccompanied baggage service. Caught@Work would also have to arrange to have the shipment picked up at baggage claim or shipped onward from the airport.
The originating airport does not necessarily have to be from LA or San Diego if the package could be shipped from San Diego to a helpful traveler elsewhere in the US or Canada.
If you decide to go with air shipping, I hear BAX Global is usually reasonably priced. It should be around $200 for your package. I’ve never used them, but my bike shop just told me my new bike will be shipped by them. The only catch is that they only ship to airports or business addresses.
Also, you could try looking into moving companies rather than package delivery companies. Sorry, I don’t know about any specific companies.