Ok, got the sound sample... thanks for reminding me about that email address. I thought I changed that forum contact email and it turns out there were 7 unread emails in there
Anyway... You are right, that sound seems intentional by some piece of hardware and your observation seems to indicate that it is occuring when the app fade starts for the cut songs. With purchased songs, I can fade start things by manipulating the raw audio sample data. With Apple Music songs, I cannot access the raw data and must manipulate the volume of the entire device. I suspect that sound is warning you that the volume has been turned down to zero. Perhaps that warning could be turned off in the settings for the exogear device or the Apple device.
If you can't find anyway to disable it, I might be able to change the fade logic to go close to, but not all the way to zero. You could test that theory by manually running your volume down to zero and see if it makes the tone and then try close to but not quite zero.