I think CO is your best bet. You would fly either via Newark or Houston... if you want to maximize your time spent in better seats, go through Houston.

Your flight from LAX to Houston would be in First class if Business isn't available. It will be noted when you book the flights.
It would be a little over $1400/person.. so x3 = $4300 total.
Now that CO is part of Skyteam, your travel can be credited to any number of Skyteam partners (Air France, Delta, NW, CO, etc). Might as well get credit for a premium fare.