I do not capture video myself, but I can tell you that if you specifically want to mute the music, but keep everything else after you have recorded the gameplay, then the only way to do it is if the Hauppauge capture device records each audio channel separately. Then you can edit / mute that specific audio channel.
For example, if you play an online game with 4 other people and all of are using headsets to talk to each other, then there would be total of 8 different audio channels (or audio sources) that can be recorded separately assuming the capture software / hardware allow you to do so. In this scenario the audio channels would be as follows:
#1 - In game dialog (if any)
#2 - Sound affects
#3 - Music
#4 - Mic #1 - You
#5 - Mic #2 - Friend #1
#6 - Mic #3 - Friend #2
#7 - Mic #4 - Friend #3
#8 - Mic #5 - Friend #4
I know that OBS Studio allows you to record each channel separately, but that is software so you will take a performance hit when recording game play. When rendering your video for upload you can simply mute channel #3 to get rid of the music. Also, say if you want to upload kid friendly content and Friend #4 is always cursing throughout the game, then you can also mute channel #8 as well.
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