Believe it or not, Byron spends so much time on the couch towards the end of this story because he is concerned about his younger brother Kenny and the way that he is acting since they returned to Flint after the terrible experiences of Birmingham and the violence and death that Kenny witnessed. Kenny found himself hiding behind the back of the couch more and more, as he believes it is the "World-Famous Watson Pet Hospital" and he, like the various pets before him, will be healed if he goes behind it and stays there long enough. However, what is clear is that he is distancing himself more and more from everybody around him. When Byron realises where Kenny is hiding, he spends a lot of time on the couch, even sleeping there, because he loves Kenny and is concerned about him. Note what he says to him after he manages to make Kenny realises he has nothing to feel guilty about:
Today is the day you check out of the World-Famous Watson Pet Hospital. Don't let me catch you back there no more. You ain't got no cause to be ashamed or scared of nothing.
The love that Byron has for Kenny is rather ironic given how nasty he was to him at the beginning of the story. Yet it seems that not only has the trip to Alabama made Kenny grow up, it has also made Byron grow up, and his spending more time on the couch is a clear sign of his love for his younger brother.