My wife regularly brings her chihuahua from back and forth between the two countries; just have a dog carrier that doesn't look like one (more like a purse) and I hope you are a girl. It works beautifully!
You can buy a small travel pet carrier and most airlines allow the pets on board as long as they are kept in the carrier and it fits under the seat. I don't know if there is an extra charge for that. I was on a flight recently with a small Chihuahua in such a travel carrier. It could have been a companion pet which must be allowed to travel with it's registered person.
However, it sounds like the OP is looking to ship the chihuahua rather than travel with it. I would suggest checking with a local vet in her area who may be familiar with shipping pets to foreign countries.
Poor pup, how many hours is a flight from California? 8? If it was travelling with someone, at least it wouldn't be alone but it's a long haul for a pup, especially since they don't recommend sedation if it's going in the cargo area during winter
My three chihuahuas would be thrown into the sea from 35,000 feet by unanimous vote of the passengers and crew!
Leaving behind many dead and wounded,and everyone deaf!
But they do love my girls,and ME Too!
oh go back and chew on your bone Tambo, that is what you do best.
I intend to bring in dogs in the future if I see fit to for my pack despite that there are many dogs here with needs. And one has won my hart already and is part of the family already.