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US Airports are not separate like most Australian airports are. International and domestic departures often leave from the same terminals, as the US does not have as strict physical controls as Australia does. Airline departures are done as you have surmised by the airlines, via the APIS system.
So your conclusions are correct. You will land at LAX Terminal 3, jump on a shuttle bus to take you to TBIT, then hang around for your departure.
This is confirmed by the LAX page on transfers under the section Inter-terminal bus past security
A bus provides connections between Terminals 2, 3 and Tom Bradley past the Security Screening Check Points.
Click here for interactive map