  The two-dimensional easy-plane SU$(3)$ magnet subjected to the transverse
field was investigated with the exact-diagonalization method. So far, as to the
$XY$ model (namely, the easy-plane SU$(2)$ magnet), the transverse-field-driven
order-disorder phase boundary has been investigated with the
exact-diagonalization method, and it was claimed that the end-point singularity
(multicriticality) at the $XX$-symmetric point does not accord with
large-$N$-theory's prediction. Aiming to reconcile the discrepancy, we extend
the internal symmetry to the easy-plane SU$(3)$ with the anisotropy parameter
$\eta$, which interpolates the isotropic ($\eta=0$) and fully anisotropic
($\eta=1$) cases smoothly. As a preliminary survey, setting $\eta=1$, we
analyze the order-disorder phase transition through resorting to the fidelity
susceptibility $\chi_F$, which exhibits a pronounced signature for the
criticality. Thereby, with $\eta$ scaled carefully, the $\chi_F$ data are cast
into the crossover-scaling formula so as to determine the crossover exponent
$\phi$, which seems to reflect the extension of the internal symmetry group to
SU(3).
