> It turns out some domestic airlines will permit you to fly with a pet on-board in a soft-side case under the seat where one might normally put a bag. This may or may not work for you, depending on how patient and large your dog is and how big your budget is.

The person to whom you replied is trying to import an animal into a foreign country. Doing that, even for pet-in-cabin and service animals, requires a certificate from the USDA. The same process as domestic flights won't work in this circumstance, unfortunately.