Benzoic acids are stronger than carbonic acid, so reaction should take place. However, both copper carbonate and copper benzoate are insoluble in water and organic solvents and both show almost the same color. And this is the main problem. Moreover, this reaction is heterogenous and must proceed slowly.
The better solution of the problem is to dissolve copper carbonate in hydrochloric or sulfuric acid (or use of soluble copper salt) and use of sodium benzoate instead of benzoic acid itself.