  We investigate the inverse moment of the $B_s$-meson light-cone distribution
amplitude (LCDA), denoted as $\lambda_{B_s}$ and defined within the heavy quark
effective theory, through the calculation of $B_s \to \eta_s$ form factors. The
presence of the $s$-quark inside the $B_s$-meson dictates a notable departure
of approximately $20\%$ in the $\lambda_{B_s}$ value compared to the
non-strange case $\lambda_{B_q}$, as computed within the QCD sum rule approach,
albeit with significant uncertainty. First, we compute the decay constant of
the $\eta_s$-meson utilizing two-point sum rules while retaining finite
$s$-quark mass contributions. Next, we constrain the parameter $\lambda_{B_s}$
by calculating $B_s \to \eta_s$ form factors within the light-cone sum rule
approach, using $B_s$-meson LCDAs, and leveraging Lattice QCD estimates at zero
momentum transfer from the HPQCD collaboration. Our findings yield
$\lambda_{B_s}$ = 480 $\pm$ 92 MeV when expressing the $B_s$-meson LCDAs in the
Exponential model, consistent with previous QCD sum rule estimate yet
exhibiting a 1.5-fold improvement in uncertainty. Furthermore, we compare the
form factor predictions, based on the extracted $\lambda_{B_s}$ value, with
earlier analyses for other channels such as $B_s \to D_s$ and $B_s \to K$.
