For me, one way of thinking through questions like this is to consider the themes or message of the story and then to pick a song based on one of those themes. One obvious theme of this excellent short story is the relationship between the two brothers, but also how the narrator, the elder brother, seeks to train and teach his younger brother to be able to do activities such as walk and then run for the sake of his own pride. Such a complicated fraternal relationship is obviously a big part of the story. A song that comes to mind straight away, which is a bit dated now, is "Brothers in Arms" by Direstraits. It captures the kind of interesting dynamic in the feelings of the narrator towards Doodle but also establishes the clear sense of overriding love that is evident at the end of the story.