Here's a couple of ideas, if you look at Google earth of the parking lot near Aviation Blvd and W 11th Street by LAX, you'll see the back of the parking lot is filled with RV's. I'm guessing that is some sort of long term parking lot for the airport for people doing just what you're doing. Looks like the lot is fenced in, but who knows how good the security could be.
A second choice is to fly out of one of the commuter airports into LAX to start your trip. For example I bet you would have a much easier time finding short term RV storage around Laughlin, NV, than you would LAX and they have commuter flights over to LAX. The money you save on parking your RV at LAX may even pay the extra for the tickets. And sometimes you find that using the commuter airports doesn't really add a lot to the price. There are all kinds of commuter airports in more RV friendly places like Palm Springs, Laughlin, maybe even Las Vegas.