Is the dog inside of the house, or inside of a fenced yard? The OP does not say.
Many service delivery folks won't deliver to my house because my dog is in the yard, inside of the fence. I've no problems with this. They either leave me a note, or use their cell phones to call me and I escort them into the yard, or make arrangements to pick up the delivery. My mailbox is located outside of the fence, so i have no problems with postal delivery.
If your dog is inside of a fenced yard and the Post Office refuses to make delivery because the dog is in the yard then the solution is simple: put up a Post Office approve mailbox outside of the fence. If your dog is inside of the house during the mail delivery and the postal worker refuses to deliver the mail, then contact the Supervisor at your local Post Office and inform him, or her, of the situation, and work out a solution.
Just some thoughts...