I boarded #2 in LAX earlier this week (with two young kids), and they called for boarding at around 9:35, very shortly after the train actually arrived from the yard. I supposed there was a two minute window where a redcap could have squeezed us in before the general call for boarding, but in my opinion, it's also equally likely that a redcap could have gotten delayed on their way to pick us up and got us there after we could have got there ourselves. Remember, a redcap's primary purpose is to assist people getting themselves and their bags to the train by moving them there on a cart. Any speed advantage in boarding is purely a side effect. It's not worth paying the tip to potentially save 2 minutes, IMHO.
There's also not a lineup in LAX. (There is a line for coach passenger check in, but that's for seat assignment, not boarding). Once the train's there, you can get on right at your car. You don't have to wait behind a bunch of coach passengers to get on first, so early boarding for sleeper passengers before coach wouldn't accomplish much since you're all going to different spots of the train.
So, I wouldn't sweat it. Use the time in LAX to get them into their pajamas or get their teeth brushed and get anything else of their nighttime routine out of the way. Then have something to read or do until they board the train. In my case, I assumed my kids would be tired and cranky, and I was trying to keep my kids in the waiting room reading quietly. But, my kids love Union Station, particularly the garden courts on the side, and kept demanding to be taken out to see the flowers and fountains. It was way easier to just do a nature walk for an hour than to try to occupy them inside. Then, once on board, they went right to sleep.