New to LAX? I'm so sorry. I feel i should prepare you with a humble native Californian opinion- LAX SUCKS. SO MUCH. 8 terminals- not all connected. (Make sure you get dropped off at the proper terminal! We had connecting flights this spring on opposite ends of LAX -not our choice, airline rebooked us on a partner airline- it took 1 hour gate to gate and we half ran where we could, but we waited for the terminal bus, which was confusing in itself come to think of it, because that would have been way too much walking. Terminals look so much closer on that map. So, pay attention to the signs when you're getting dropped off. I know it sounds like a no-brainer but huge airport, early morning, tired, kid, family, tickets- this is a little thing you don't want to get wrong.) Also, LAX is always under serious construction. And so much traffic. (But prob not bad that time of morning, so YAY!)
Now that you're super-stoked for your LAX experience- to business! I wouldn't do this- but I have only flown in and out of LAX a couple times this last year. And am practically paranoid about having enough time there at this point. Mainly because LAX TSA has been awful the last couple times I went through. Have TSA Pre-Check to solve that problem? ROFL. Those lanes are only open every 6th Monday of the month. And hours are limited. But seriously, don't count on it if you've got it.
Personally I'd spring for a private car that will have the car seat you need, just dealing with your toddler at LAX will be stress enough- don't add to it by worrying you'll be late or trying to race through LAX to your gate. DISers are a good resource if you need suggestions. (Sorry I can't help- no kids.)