I can understand the challenge and you are probably aware that Audacity has a noise removal tool. If you can isolate the barking and copy that you could possible remove the barking from the recording. However, as several have said here, you would probably remove frequencies that are an essential part of the music. The odds at removing only the barking are very slim. If you primarily want a clean version of the song, you might give us the name and any other information and see what happens.