You best bet IMHO is to head to T4 and the American Airlines Flagship lounge (which you have access to at the very least for being inbound to LAX in First Class on a oneworld flight of over five hours and connecting to another oneworld flight). You could go back though security at TBIT but IMHO, while the Qantas First lounge there is fine, the Flagship lounge is better (certainly in the mornings) and more convenient for your flight to JFK.
There you can relax, enjoy a shower and take it easy while waiting for your flight to New York.
Moreover, if you are in First on a three class AA flight to JFK (A321B aircraft) then you are entitled to use the T4 Flagship entry with porter assistance and expedited security. That entry is a small red velvet appointed door manned by staff wearing cap and coat tails. (
Flagship Check-In)