From the SoundSource manual:
SoundSource is what’s known as a “menu bar application”.
From that same page:
Finally, SoundSource can be accessed and controlled via the keyboard. When the configurable global keyboard shortcut is pressed, SoundSource’s main window will come forward and receive focus, enabling you to see your current settings, and adjust them.
So if that hot key works when sent from KM, that would seem to be the best method.
Of course, that depends on how you want to interact with SoundSource. It may be that you can do everything you want without even popping its window. Take a look at the Shortcuts actions available -- perhaps you can write a Shortcut to do what you want, and run the Shortcut from KM?