As a matter of law, you must collect your baggage at LAX because of CBP. But your baggage should be able to tag all the way to PHX without issue, especially you are an AA elite.
In term of refund, it will be difficult as it is not AA collecting the money, but CX/KA. So you may have some difficulty arguing about the refund.
You should simply let them offload your baggage and deal with it at LAX, as I am sure an AA GA will check the baggage for free.
BTW - it is not that far from TBIT to T4/T5.