You will find that no one coming to live here has imported a bird. I wanted to import a tortoise a few years ago but gave up. The paperwork, requirements such as quarantine and veterinary services and the cumulative expense was far beyond what made practical and financial sense.
Birds are particularly problematic in the eyes of the authorities as they can potentially escape their cage and fly off into the ecosystem. Dogs and cats can move in and out of the DR pretty easily but other live animals are for the most part discouraged unless for the purposes of a zoo type operation.
Talk to your bird and see if he/she will train a dog/cat to provide the same service. A new home for your bird in the DR is not going to work out, sorry..