Wow...this will be a fun day.
First, taking a train from San Diego station to the LA Union Station, then getting from there to LAX is one option. Take a look at the Surfliner (the train line that runs San Diego to LA...the station is about 4 or 5 blocks from the pier)schedules...that might work. I don't know the current best way to get from the train station to LAX.
The other way would be renting a car. So getting over to San Diego airport car rental facility, and driving up 5 and 405. 405 can be a problem any day of the week, any time.
Yes, I think you can make it. But what a complex hassle. I would rent the car.