If you have the correct manual with the schematic, please post it. Based on the DX-150 manual -
The additional 160 mA load is increasing ripple out of the power supply bulk filter capacitor to the point that it is getting into the audio power amplifier at an audible level. There are no differential stages, current sources, or any other circuit techniques that reduce power supply sensitivity. The radio circuits are powered by a voltage regulator that reduces the ripple seen by those circuits, but the amplifier section is not.
The bulk filter cap is C64, 2000 uF / 25 V. Double it with an additional 2200 uF, or replace it with 4700 uF.
In the audio amp, it looks like C60 is 1 uF / 16 V. Increase to at least 10 uF, preferably 100 uF, whatever will fit.
If these changes do not improve things, then it the ripple is getting into the audio through the audio amp output stage. Nothing you can do there.
ak