There's a function in Audacity which lets you select a wave (i.e. a clean sample of the "noise" you want to remove) and then it tries to remove it from the rest of the sound.

It's a bit hit and miss.

How would you characterise the winding noise? Is it constant or periodic?

You could try sampling just the winding noise, then inverting the wave and mixing it back with the melody + noise... again, would be very hit and miss.

Is there any way you could find a "wind up duck" preset on some synth and just copy the melody?

P.S. You're a lunatic.