I know what you're dealing with. This is just a main part of the game.
You could get a fur texture to get rid of this. I'm pretty sure if you look hard enough you will find a texture that looks like the feline texture but without the problem you have.
Hope this helped. I will try to find a similar fur texture for you.
It isn't a problem with the actual fur texture, it's the mask with the eyes and mouth. There's a lot of headmasks alone, it can get confusing