Solving carbon black
By far the safest solution is to physically abrade the surface until you have worn away everything that isn't copper. This is slow and requires some sort of grinding agent but has the upside that you are not likely to injure yourself.

There is a chemical solution that will work but it requires access to some dangerous chemicals and is extremely hazardous even in a well-equipped lab. It is insane to try anywhere else as you won't have adequate safety equipment.

The best way to remove carbon black from anything is to use a mixture of concentrated sulfuric acid and hydrogen peroxide. This is often used (but also often highly discouraged because of the danger involved) in laboratories to clean sintered glass filters contaminated with things like activated carbon. In the case of a copper bowl, the solution will remove the carbon but will also etch the copper surface (it is sometimes used as an etchant to remove copper from printed circuit boards). 

The mixture of sulfuric acid and hydrogen peroxide is sometimes known as Piranha solution of reasons that are obvious when you have seen it work.

I would not encourage anyone without good chemical safety training and access to very good safety equipment to work with the mixture. And even then they should work only with small amounts and in a fume hood behind a shield and while wearing eye protection, strong gloves, and a fireproof coat.