Welcome to FlyerTalk!
COPA will honor what Continental's policies are pretty much. But I'd worry a bit about Chile - be sure you have a veterinary health certificate, rabies certificate - perhaps translated into Spanish to boot.
thing IMO would to check with breeders' directories, contact Chilean breeders and ask them for information and recommendations, with plenty of time before you travel. Ditto contacting prospective hotels by e-mail before booking.
Remember, there are many countries around the world where travel with pets is strictly unusual; food and water, changes, etc. might be stressful to them, and diseases like pinworm may be common. It's better to be prepared.