If set $C$ is recursively enumerable and $B$ is Recursive, and if $B-C$ is recursively enumerable then is $C$ recursive or not?
The argument is wrong.

Your argument is: $B\cap \bar C$ is r.e., $B$ is recursive, so $\bar C$ has to be non recursive, otherwise $B \cap \bar C$ would be recursive -- contradiction.

However, there is no contradiction: $B\cap \bar C$ can be both r.e. and recursive. This is equivalent to $B\cap \bar C$ being recursive. There is no hypothesis contradicting that.

Indeed, take $C$ to be the halting problem (r.e., and not recursive). Take $B=\emptyset$ which is recursive. Then $B\cap\bar C = \emptyset$ which is (recursive and) r.e. . So all the hypotheses are satisfied, but $C$ is not recursive.