I think you're barking up the wrong tree with the inductor saturation idea.
The controller doesn't switch the inductor, it switches a MOSFET. The inductor current is monitored, either by being sensed directly or by being reconstructed by a separate circuit, and when it reaches a predefined threshold, the MOSFET is switched. There's no sense in which the same MOSFET can become "difficult" to switch.
What can become "difficult" is determining the proper time to switch, in a noisy environment.
In an ideal world, monitoring (or, in some circuits, modelling) the inductor current results in perfectly uniform switching cycles, so the only frequency components present will be the fundamental (say, 500kHz or whatever), plus higher harmonics. If the inductor starts to saturate, it'll heat up, the supply becomes less efficient, and the output voltage will sag. The controller compensates by increasing the PWM duty cycle, which may work up to a point, but the circuit is now working beyond its realistic upper limit. It still shouldn't be making audible noise, though.
What happens in a real circuit is that switching noise from the MOSFET couples into the voltage feedback and/or current sense nodes. This results in random, chaotic variation in the PWM duty cycle. It still averages out to the proper value over time, thanks to the closed loop control, but it means there are audible components in the frequency spectrum of the switching waveform.
The noise scales with current, and so, therefore, does the magnitude of the chaotic variation.
I suspect that changing inductor changes the noise because of each part's mechanical characteristics, but it's not really making any difference to the real reason why the noise exists in the first place.