I am assuming you will be burying the dog’s ashes. If you would like to pour a little holy water and ask the Lord to please bless you and take care of your dog, or express your confidence in God that He has already taken wonderful care of your dog, that would be fine. As someone else said, the dog does not need any sort of cleansing of sin, as dogs do not sin and your dog has already been cared for by the Almighty, so any grace involved would likely be coming back to you and your family with God giving you healing and strength as you mourn the loss of your dog.
No need to bury holy water and especially not in a container.