You could use the stencil buffer:
- make sure the hat is drawn before the fur (for eg, create the hat object before the fur ball)
- draw the hat in the stencil buffer with a 1 value
- draw the fur ball with a comparison function “!= 1” so that it does not overwrite the hat
However, you will see that depending on the view, you can still see the tip of the sphere, and you would need to enable the stencil writing only for the front faces, not the ones on the back. In your case, the hat has some thick edges and you may not need to look from behind, so maybe it would work.
I think a clipping plane that would clip the tip of the fur ball would work, however:
Of course, you will need to adjust the plane if the fur ball is rotating.