Can I ask a really dumb question, why do you have to get your ticket? Can you not collect it from the transfer desk and therefore if you have your ticket number have Air Cal check your bags all the way to LAX? That way you simply use the transit facilities at SYD airport.

Or is someone else who is not travelling with you to LAX holding a physical paper ticket for you? If so, I hope you don't have issues with Australian Immigration wanting to see a ticket for your departure if you are not an AU or NZ citizen.