We have shipped dogs oversees and they traveled cargo in the same plane as us. There are shipping pet services where you drop of the dogs and they handle everything. They will probably keep them overnight. Call your airline and tell them what you need and who sells the crates.
Our dogs, about 10 years ago cost $175.00 like that and were in cargo in our plane. We provided the crates, airline approved. I had to arrange all that with the cargo department of the airline.
I would turn the world inside out to take them with me. It sounds like you are doing just that.
Deja, GSD, sable WL, female, born 11 / 2013
Cam, Rough Collie, sable, male, born 8/2015