I believe there is no baggage storage at the airport itself.
This is true, at least it was 6 months ago when I was there.
There are luggage storage facilities but they are located off-site for security reasons.
You can phone them (and I think pre arrange) for them to come and pick it up and drop it back for you.
In the end I didn't need to, but here is one such service, I'm sure there's more;
LAX Luggage Storage.com - For your baggage storage lockers at the LAX airport
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I would check my bag at Marriott West Century and pay a suitable tip on pick up. Their baggage room can handle it. Catch their free shuttle.