As you can see that pixel has two different assigned colors. It flickers a lot if moving with the tool. Thus I guess, the texture needs to be fixed.

That happens any time you have more than one texture in layers on the model. Unfortunately, that's the only way I could get all of the textures for the tools without making redundant textures.

Do I understand it right that, that some handle and head textures provide the pixel and some don't and that only happens if both provide it?

The Grip texture is another layer on the texture; when multiple layers on the same model cover the same pixel, this happens. The Grip and Handle texture have different colors on the same pixel, causing this.

How about using a stencil buffer to resolve that z-fighting?