Most likely your grand father's dog was used by the USMC during WWII.
All USMC WWII records were recently located (found) at Camp Lejune, NC and transferred to the "National Archives" in Washington, DC.
You can contact the National Archives with the information you have and they can provided a complete copy of the records about the service of any of the USMC WWII Dobermans.
I'm not certain about how muh information they need, but call them and find out.
I'm certain that there is a charge for copies of the records, but I don't know how much.
Mike Limish provided the information about the discovery and transfer of the records during a conversation yesterday at Fort Benning, GA.