I would definitely opt for flying him. A flight is a lot shorter than a Pet transporter. Furthermore, you have said yourself that he is really afraid of other people, and jump out at any noise. Pet Transporters involve people, flying your dog does not. Airline does not open the cage, which they usually lock with plastic ties because this is how dogs are lost. By opening the cage to take your frighten dog for a walk, a Pet Carryer could lose your dog.
Most dog do sleep in the plane ride just as they tend to sleep in a car ride. Although there is no doubt that the experience can be a difficult one, it is by far a much shorter one, with no immediate threat such as people approaching, talking, other dogs being taken in and out, crate being opened... By the time your dog worries a little, falls asleep, her you will be, greeting him at the other end of the trip.
You have 3 months to prepare. This is what I would do:
1- go to your vet, explain the situation, and ask that your dog be put on a psychotherapy (antidepressant) drug 1 to 2 months prior to the trip. They work well for humnan and they work well for dogs.
2- try to fly direct. If this is not possible, avoid changinf airline even if it means for you to have to drive to an airport a little further away from your homecity.
3- if you do not have one, get your crate right away and teach your dog to stay in the crate a little time each day, wqithout closing the door at first.
4- DO NOT drug your dog before the flight. This is actually now illegal. Drugs designed to make the animal sleepy lower the heart rate which put a dog in danger. It also lower the body temperature, and the dog needs his warmth to be comfortable during the flight. Compartment for live animal is not heated and is about at 5 to 10 degree celsius depending of the season.
5- rather than cargo, put your dog as an accompanied luggage. This mean that you will fly in the same plane as your dog, will be the one putting him at the belt as you check in, and will be able to get him immediately as you get your luggage. This also allow you to ask the airhostess to double check that your dog is in the plane as soon as you board. If you need to change flights, you can do that for each flight.
6- finally, don't be nervous. People travel with their dogs all the time and although you hear of the occasional story, there is far less chance for something to happen to your dog while flying than there is for the Pet transport to have a road accident. Further, if you are anxious, so will be your dog. Moving is an exciting time, especially to California. I'm jalous already. So make it a happy time. Talk happily to your dog about it, especially when you will start packing.
Remember that I am going in April to a Seminar who will be dealing with the issue of travelling with your dog. So if you have any particular question you feel no one has answered to your satisfaction, please don't hesitate to let me know. I will be happy to ask the specialist on your behalf.
Good luck and let us know what you decide.