  We consider the feasibility of interpreting the $R_{K^{(*)}}$ anomaly in the
colored Zee-Babu model. The model generates neutrino masses at two loops with
the help of a scalar leptoquark $S\sim(3,3,-\frac{1}{3})$ and a scalar diquark
$\omega\sim(6,1,-\frac{2}{3})$, and contributes to the transition $b\to
s\ell\ell$ via the exchange of a leptoquark $S$ at tree level. Under
constraints from lepton flavor violating (LFV) and flavor changing neutral
current (FCNC) processes, and direct collider searches for heavy particles, we
acquire certain parameter space that can accommodate the $R_{K^{(*)}}$ anomaly
for both normal (NH) and inverted (IH) hierarchies of neutrino masses. We
further examine the LFV decays of the $B$ meson, and find its strong
correlation with the neutrino mass hierarchy, i.e., $\text{Br}(B^+ \to K^+
\mu^\pm\tau^\mp)\gtrsim\text{Br}(B^+ \to K^+ \mu^\pm e^\mp)\approx\text{Br}(B^+
\to K^+ \tau^\pm e^\mp)$ for NH, while $\text{Br}(B^+ \to K^+ \mu^\pm
\tau^\mp)\ll\text{Br}(B^+ \to K^+ \mu^\pm e^\mp)\approx\text{Br}(B^+ \to K^+
\tau^\pm e^\mp)$ for IH. Among these decays, only $B^+ \to K^+ \mu^\pm e^\mp$
in the case of NH is promising at the LHCb RUN II, while for IH all LFV decays
are hard to detect in the near future.
