If the doctors were listed in the sentence (e.g. Dr. Smith, Dr. Jones, and Dr. Why), then the verb would need to be plural. In the sentence as written, the clause about Dr. Jones and Dr. Why is more similar to a parenthetical aside -- it's not really part of the subject. Therefore, the verb should be singular to match the singular subject (Dr. Smith).
If you think you might be able to get away with it, you could also rephrase the sentence to "Dr. Smith and his colleagues, Dr. Jones and Dr. Why, have been working on purple cat research." That makes the subject "Dr. Smith and his colleagues" and the part with their actual names is a clarifying aside separate from the subject. It keeps the emphasis on Dr. Smith, but it might read a little better.