Can a perturbation of a matrix product always be represented as product of perturbations of its factor matrices?

The condition you want is exactly that the matrix multiplication map be locally open at the pair $(B,C)$. This is the topic of the recent paper Where is matrix multiplication locally open? by Behrends. The paper contains a complete characterization in Theorem 2.5. According to that theorem, if we let $s$ and $t$ be the ranks of $B$ and $C$, respectively, and let $t = t_1 + t_2$ where $t_1 = \dim (\operatorname{range} C \cap \ker B)$, then matrix multiplication is open at $(B,C)$ iff $t_2\leq k-m$ or $n-t_1\leq k-s$.

In numerical analysis lingo, you are more or less asking if matrix multiplication is backward stable. The answer seems to be no: see Section 3.5 of Higham, Accuracy and stability of numerical algorithms. I am unable to locate an explicit counterexample quickly, though.