The dog may also be a metaphore for his job and the firm (or at least the team) that he is running. Rogers is a compassionate character who cares about the work, but is not afraid to take hard decisions. His speech at the end shows that he recognises what he has done. The film focuses on his moral choices.
The dog is a selfish indulgence - he spends a fortune on it. It is sick with cancer, as is the business he works for. It is living on borrowed time, being kept alive by money (like the business). In the end, when it is clear the music has stopped, he puts the dog down, destroys his job with a quick death (he puts it down), destroying his relationship with his clients. He has no control over either event. In the end, he accepts what has happened, buries the dog, mentally buries the events of that day (he decides not to quit) and moves on.