Uber Black shouldn't be that expensive with a 12 mile ride. I'd just use that and get curb side pick up.

Alternatively, use the bus... I'm not kidding, I think you live close enough to LAX that this should be an option. When my son was younger and still using car seats, we just walked 1/2 mile to the bus stop on Lincoln Blvd and took the Big Blue Bus to the airport and transfer to the LAX shuttle to the terminal. We must have done this 6 or 7 times and it was very easy and the most painless way to travel with very young kids. Return trip is the same, just need to time your arrival flight while bus is still running frequently. Never had to worry about car seats.

I don't know which direction you live but 12 miles will put you roughly in Westwood or North Santa Monica. If you live in the Westside, use Big Blue Bus or Culver City Bus to LAX. Both very easy to do and both runs fairly frequently throughout the day. The only time bus option wouldn't really work is early morning or late night.

We went back to using Uber once my son was in booster seats (he is 7 now) so it was much easier to install and take out of the back seat.