List of transporters I've seen...
CEVA
Patricks
Toll Auto
Adelaide Car Express
AutoTrans
GJ Roberts
Probably a few more that I can't remember now. Problem is people are always sending stuff TO Perth, so there is no backload as such. Backloads go East
Also, if you are going to be here to pick it, the Indian Pacific does unaccompanied vehicles as well.
PLEASE PLEASE PLEASE clean your car before you send it over. All used vehicles sent over are subject to Quarantine inspection and cleaning upon arrival in WA. There should be NO dirt, seeds, plant material etc on the car, in the radiator, in the carpet etc etc or else it will need to be sent for specialized cleaning when it gets here which is not cheap. Also the Quarantine inspectors will go through your personal effects to make sure there is no Quarantine Risk Material in there like plants, seeds, soil, fruit and veg, untreated animal hides, gardening equipment, nuts in the shell, honey, rocks, firewood etc etc. Whoever you decide to send it through, make sure they are aware of Quarantine or you may be in for a rude shock.
Best of luck.
Keels (Quarantine Inspector for Quarantine WA)