You shouldn't have to check them again at LAX if you let the agent in AUS know that you have another flight from LAX-HKG-SIN. Just give them your flight numbers and they should be able to find your flight and even check you in at AUS all the way through. You might want to check in at the CX counter in LAX to get CX boarding passes. They can also check your baggage tags to see that it is checked all the way through to SIN.