OP: Don't sweat it. Since you are a DL PM on an international flight, you will get access to the SkyClub. You arrive in the morning, which gives you plenty of rebooking options if you misconnect. Just relax in the SkyClub and freshen up (they even have showers there) after such a long flight.

Oops, I just noticed you booked two separate tickets. This will complicate matters should you misconnect. As soon as you land at LAX (i.e., still on the runway), if there appears to be ANY possibility of misconnecting, get on the phone with SMS and ask to SDC onto a later flight for your other ticket. You have no protection for your second ticket if your SYD-LAX flight gets jacked up.