During the three weeks, may parents put on music that helps their children sleep? Perhaps there is a difference between before Rosh Chodesh Av and after Rosh Chodesh Av.
Generally, if the music is not for one's specific enjoyment, you can listen to the music. In your scenario, you are not the one listening to the music, anyway - your child is. And, s/he is not even listening for the enjoyment of the music, itself, anyway. Plus, there may even be more leniencies for a child below bar / bat Mitzvah, anyway.
In summary, I wouldn't see a problem, personally, but confirm with your Rav.
See this article for details.