I flown to LAX for Disney probably about 5-6 times in the last 5 years (most of them in the last 2).
The worst experience we had was in the sdummer of '01 (this of course before security got really bad). We were running late because of traffic (lesson learned - leave Anaheim early!). I dropped off my wife and the kids while I went to return the car. Well, I thought I did. We wre flying American, and at least at that time, American was at 2 diffrent terminals - one marked "American" and one marked "American - Cities." Silly me, I thought Chicago was a city! So we loaded the luggage back in the car and made the short drive to "American" on the other side of the international terminal.
We got all of our boarding passes (the days before online check in), and I went off to drop of the car. It was not the easiest to find
Avis - I must say that of all the dozen or so airports I've rented cars from, LAX was the hardest to find the return area.
I raced back to the gate, only to find that the plane had pulled back exactly on time, some two minutes ago. So I asked the gate agent to get me on another flight. "We show you boarded," she said. Producing my boarding pass, I found that hard to believe! Some stressful hours later, I was flying home without my family (but with another pasenger who thougfht Chicago was a city).
Since then, I've flown out of LAX alone or with my wife, but never with the kids.
That being said - you can survive - even with a toddler! Getting out of the airport isn't bad (sorry, O'Hare is much nicer and easier, albeit bigger. That's part of LAX's problem, it should be bigger, but it looks decidedly like not a penny has been spent improving it since 1975). The Anaheim Supershuttle always seems to be the last to arrive, but unless it's raining, the wait should be tolerable.
The last time I used baggage claim, the area seemed very unguarded, which I thought odd.
Getting back is a bit more of a challenge. I'd say leave Anahiem about 3.5 hours before your flight at least (especially if during the middle of the day - then 4-5 hours is not out of the question).
The security area was really given very little room (at least at the Southwest terminal - where I've flown the last few years). We had to wait once in a 2 block long line outside. We were still through in about 40 minutes, it's just there was no room inside for people to line up. Also, they would check your pass at the bottom of an elevator, and then have the screening right after the top. This lead to a line at the very top of the escalator with no room for those coming up!
If you plan ahead you'll be fine. I'd try to carry on flying into LAX, but I'd check as much as I could flying out (to avoid having to carry a lot through security). You can't leave too much time for the trip back - better you wait around the airport than miss your flight.
I've also found I can get similar rates in San Diego. While this is 90 miles away, there are lots of attractions (Sea World, San Diego Zoo, Wild Animal Park,
Legoland) between San Diego and Anahiem, the airport is wonderful - big but not crowded - a breeze to get in and out of. The drive is scenic and generally without much traffic.
But you'll do OK - it's those that don't prepare for LAX that get messed up!