QF F lounges at MEL and SYD are amongst the finest lounges in the world, far better than any F lounge at LAX. Unfortunately for you, admission is based on the class flown for the departing QF flight (which is J). You can ask to have the rules stretched, showing your F arrival BP, and have a good chance to get into the SYD F lounge (you can't get into the MEL F lounge as you aren't departing internationally, but the domestic J lounge is nice enough, putting the AS Boardroom to shame). Your routing significantly devalues F, fwiw, and flying J LAX-SYD-AKL would probably have been a superior experience to F LAX-MEL and then J MEL-SYD-AKL (too bad MEL-AKL wasn't available).

At LAX I normally rent a car and drive to Manhattan Beach -- much nicer than any lounge. Another alternative is a day room at a hotel near LAX (more useful if you need sleep). Another great outing is to drive up PCH (Pacific Coast Hwy) to the Ghetty Villa if you want culture, a nice view plus some OK food.