Koans are given to you personally by your teacher. You can't just pick up someone else's koan and think it will do you any good.
The Buddha did not teach Buddha nature. He did not teach with koans. He instead spoke so that he would be understood.
FWIW. The dog in your story will die one day and might then (depending on its kamma) be reborn as a human or into a higher realm and will then be able to achieve nibbana. While it is still a dog it will not be able to practice the Noble Eightfold Path and will not become an Arahant.
Learn more at https://suttacentral.net/
(With metta)