Long distance pet relocation

Me and my spouse would like to move from Europe to North America some time in the future. Of course there are enough problems with that already, but let's just talk about the pet relocation for now.

We have a small dog (10 kg/22 lbs) that is too big to be in the plane's cabin. The requirements for that are ridiculous because it doesn't allow for anything larger than a tiny chihuahua. The cargo section is not an ideal place for any animal, especially not a dog. There are a lot of horror stories of dogs dying there despite being checked as healthy by a vet and I can understand why. Of course the majority of them are fine, but if your dog is prone to anxiety (and mine is), there's a high chance of dehydration, self injury etc.

So I'd like to hear any personal experience you guys have, whether positive or negative, especially if it was long distance. Do you recommend any specific airline or pet relocation service? I don't want to say that cost is not an issue, because it is, but we'd scrape by whatever we'd need to as long as our dog comes out alive and healthy.