It seems that the chirp noise is coincident with changes in the stereo text readout. I.e. it seems that when the Dice unit updates the screen text, that signal interferes with the audio signal to produce the "chirp" I'm hearing.
That would explain why Aux mode and BT mode (static text readouts in both) have no chirping issues. Also explains why changing tracks (via the steering wheel or stereo controls) causes a similar chirp noise.
So open question to forum members: is there a way to shield the audio cables from the data cables as they run from my glovebox to the trunk? I know they are all currently bundled together in the black sheathing from Dice. But if I were to open that, is there some material I could use to separate the audio cables and the data cables to reduce this issue? (Though I guess it could all be for naught if the cable that BMW runs from the trunk location back up to the head unit is as poorly audio shielded as I'm assuming the Dice cable is. In fact, I guess I have no reason to suspect the interference is coming into the system on the Dice->Trunk leg of the journey rather than the Trunk->headunit leg.)