Could be a bad rear differential.


Symptoms are classic- under load it disappears, under load (part throttle) it goes away.


When the fluid is cold, no noise- as it heats up and the viscosity lowers, noisy.


If you search here and on bimmerfest for 'bad rear differential', you should find others that had issues- you can try and match up your symptoms. A DEFINITIVE diagnostic test will be hard, but you should be able to narrow it down.

GL!