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.
Related answer: judaism.stackexchange.com/a/1050– FredJul 15, 2014 at 19:40
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.