Squelching noise from back wheel
It sounds from the audio like the tyre is rubbing somewhere. The sound is certainly related to the wheel because the frequency speeds up and slows down with bike speed, not pedalling. 

You say the tyre isn't rubbing, but the sound doesn't happen when bike is unloaded, so perhaps when it is fully loaded, the frame is flexing enough to cause a rub - is there any point like behind the downtube or close to a mudguard or chain stay where the tyre is close to something? Then when it is loaded up it is able to rub.

It's also making the noise once per revolution of something (the wheel) so something is out of balance. Maybe the wheel isn't true, radially or laterally, or the tyre is bulging at one point on its circumference.