Kik is not a stock app so it does not integrate into the os like other stock apps.
If you put the phone on silent, you won't have the stock app interrupt the music. I use bt streaming to the car hu and when the phone is on silent, nothing interrupts the music. This is my preference.
I don't think there is a way to have the stock app sound play whilst the music is playing. You either hear the notification sound, or not.