  We study the local equivalence problem for real-analytic
($\mathcal{C}^\omega$) hypersurfaces $M^5 \subset \mathbb{C}^3$ which, in
coordinates $(z_1, z_2, w) \in \mathbb{C}^3$ with $w = u+i\, v$, are rigid: \[
u \,=\, F\big(z_1,z_2,\overline{z}_1,\overline{z}_2\big), \] with $F$
independent of $v$. Specifically, we study the group ${\sf Hol}_{\sf rigid}(M)$
of rigid local biholomorphic transformations of the form: \[
\big(z_1,z_2,w\big) \longmapsto \Big( f_1(z_1,z_2), f_2(z_1,z_2), a\,w +
g(z_1,z_2) \Big), \] where $a \in \mathbb{R} \backslash \{0\}$ and
$\frac{D(f_1,f_2)}{D(z_1,z_2)} \neq 0$, which preserve rigidity of
hypersurfaces.
  After performing a Cartan-type reduction to an appropriate $\{e\}$-structure,
we find exactly two primary invariants $I_0$ and $V_0$, which we express
explicitly in terms of the $5$-jet of the graphing function $F$ of $M$. The
identical vanishing $0 \equiv I_0 \big( J^5F \big) \equiv V_0 \big( J^5F \big)$
then provides a necessary and sufficient condition for $M$ to be locally
rigidly-biholomorphic to the known model hypersurface: \[ M_{\sf LC} \colon \ \
\ \ \ u \,=\, \frac{z_1\,\overline{z}_1 +\frac{1}{2}\,z_1^2\overline{z}_2
+\frac{1}{2}\,\overline{z}_1^2z_2}{ 1-z_2\overline{z}_2}. \] We establish that
$\dim\, {\sf Hol}_{\sf rigid} (M) \leq 7 = \dim\, {\sf Hol}_{\sf rigid} \big(
M_{\sf LC} \big)$ always.
  If one of these two primary invariants $I_0 \not\equiv 0$ or $V_0 \not\equiv
0$ does not vanish identically, we show that this rigid equivalence problem
between rigid hypersurfaces reduces to an equivalence problem for a certain
$5$-dimensional $\{e\}$-structure on $M$.
