It feels a little empty to do fights in Baston without there being any music to listen to. While I could just use YouTube to play music in the background, it still doesn't feel quite right.
So, is there a way to add music ingame or is it not possible?

There is no way to add music.
But it's listed in my insanely huge to-do list.

Just continue to listen to music with something else in the background, for now 😁