I did not expect operating from batteries to make that much difference.
Is the problem low power supply rejection or a flaw in the circuit or layout?
The internal diagram of the integrated amplifier shows that the bias is regulated but I wonder if the noise varies with supply voltage. It has a pretty wide supply voltage range.
When hearing the direct comparison between battery operated and main powered, there is audible hum. Since the speakers are not good at these frequency, the little hum might make up quite some µVs. Still the question if poor layout / cabling or the chip itself.
I would not expect much variation of the noise with supply unless one goes right to the edge.
I guess I was not listening carefully enough. The hum could certainly be from low power supply rejection and that would explain the multimeter readings. Batteries would of course not suffer from that.
400 microvolts and 50dB of power supply rejection yields a very reasonable 120 millivolts of power supply ripple.