So...

I was quoted about $170 to ship the bike to a FedEx centre near LAX and hold it, to which I would obviously then have had to go and pick it up.

I looked around for a few other options and eventually used BikeFlights to ship the bike to LAX Luggage Storage. They held it overnight after it arrived and delivered it out to the airport for me.

Total cost was about $150 - $100 for two-day shipping and $50 for the receiving/holding/delivering. Could have been about $50 cheaper if I'd chosen the 2-day surface shipping option, but I was a bit worried about the possibility of problems, so I went with air instead.