Don't delete your pet! If you do, you'll lose all of the data you've accumulated since you had your first Pod.
Instead, you can select the Pet whose Pod you need to swap for a new one. Then, click on the Pod button. Once you get to the Pod Details screen, you can click the three dots in the top right corner and you'll select "Swap Pod" from the options.
This will allow you to swap a new Pod into your pet's profile while keeping all of the information on your pet's profile!