Do I understood that you are creating a quilted top that will be assembled to make a dog bed? If my understanding is correct then I see why you would not want a backing.
Depending on your batting you do not have to have a a backing fabric. If you batting is fluffy (like a wool blend or some polyesters battings) then the batting may catch on your feed dogs. However, if you are using a cotton or cotton/blend batting then you probably will not have a problem quilting the top without a backing.