I have worked with both domestic and wild animals and do believe that many animals including dogs grieve. Letting the animal have some time with their deceased friend appears to help this process in that they are less likely to then search for the friend. I think if you bring your dog home in a blanket and then give your remaining dog some time with the body, then that is probably the best way to do it. And then you will also be able to bury him as you wanted to do. So sorry about your little one. It is so sad to see them get old or sick.