All requests for transport belong in the Animals Needing Rides section, regardless of the length of the transport. You would however need to break it up into segments and find several pilots to complete the mission. (Unless someone happens to be heading that way already).
The commercial volunteer section you refer to is simply for those people to post their availability. I suggest you read through their posts to see if anyone there might fit your need and contact them by PM or email. I do know there are some pilots & attendants who can take your dog in the cabin while on duty or on their days off.
You might also want to check with PetAirways as they fly from Chicago to LA and all dogs are in the cabin. http://www.petairways.com/