Cathay Pacific flies DPS-HKG-LAX. You could then fly LAX-ORD on a CX codeshare with AA or book a regular AA flight yourself. AA also flies ORD-MCO.

However, CX does state the following on their website:

Dogs and Cats in Transit
You need to apply for an Import Permit for your pet if you fly to Hong Kong, even for transit. For more details on application procedures as well as import requirements, please visit the Agriculture, Fisheries and Conservation Department's website.

Pets in Cabin
With the exclusion of service dogs, no pet or animal of any description will be permitted in our passenger cabin. Pets such as dogs, cats and birds will be crated and transported in the cargo compartment. Please check with the local government office for all documentation required (ie health certificates and travel permits) if you plan to fly with your pet.