No need for that for baggage allowance, IME. Just show them the AC *G card & it's good to go although it will require a human for check-in since the kiosk won't know about the *G status. However, the above can always be used as a back up plan if an overzealous agent tries to deny you the benefit.
LAX
You may encounter difficulties in practice as some airline check in staff would insist that in order to check in additional baggage you would have to use the card with *G logo on it. Last time departing from SYD on CA, I tried to use my NZ*G to check in some extra bags but wanted to credit the miles to A3. The check in agent employed by QF refused to do that even after checking with a CA agent near her. She even warned me that if I changed my number I would be charged excess baggage fee. I didn't listen to her and asked her to enter the NZ number anyway and had it changed to A3 at the CA service desk at the end of the row. No issue after whatsoever.