Unfortunately it will always really stand out. Obviously if you have a light colour background then it will be more noticeable than a dark background, so black will make it less noticeable - but black isn't carbon, so it will still be quite noticeable.

Sometimes I prefer to cut a few individual tows of carbon fibre at about the same size as the breakthrough, and then use them to cover the area. This obviously still stands out because they are messy strands of fibre rather than nicely woven carbon, but because it is still carbon and has the same black/goldish flip as the surrounding area, it can be less noticeable than a black dot. 

Unfortunately, if noticeable isn't acceptable, then reskinning is the only option.

I skinned a piece with carbon fiber and when sanding the epoxy resin I ended up reaching the carbon fiber and damaging it a little.

Are there any techniques to disguise this detail so that you don't need to do another skinning?

Would pigmented black clearcoat help?