Okay here is the best I could do for you. After many links and various tries I identified the music engineer of the show's episode who is responsible for music selection for the show and other shows. His contact information is right at the top it you want to ask him directly.
Chris Tergesen - IMDb
Also two compilation albums were put out with the music played on the show. House Full Of Love: The Music of the Cosby Show was released in 1986 and Total Happiness (Music from the Bill Cosby Show Vol II).
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