I ran into the same issues flying my dogs down here. They are both larger dogs, and with the crate, were going to cost a fortune on American Airlines. Fortunately for me, my US home is only four hours drive from Atlanta, so I was able to get the dogs here on a direct flight with me from Atlanta to Santo Domingo with Delta. It was a positive experience. American Airlines, even from Miami, was wanting to charge cargo rates at the tune of $2000, whereas Delta only charged $200 per dog (total $400).
This time of year, you probably are not going to run into heat restrictions. And flying all the way to Seattle is going to be an even bigger issue because that means multiple flights. Is there anyone with whom you can leave the dog until January or February, when the heat restrictions are not as bad? And can you drive to Atlanta from Seattle?
You might want to take a look at pettravel.com. The site has some great information that really helped me out when I was in the process of bringing my dogs here to the DR.
(And if bad comes to worse, I have the space to babysit a dog or two until the winter when flight restrictions aren't as bad.)