  The link between the metamagnetic transition and novel spin-triplet
superconductivity of UTe$_2$ was discussed thermodynamically through
magnetostriction measurements in a pulsed-magnetic field. We revealed a
discontinuous magnetostriction across the metamagnetic transition at
$\mu_0H_{\rm m}\approx 35$~T for the applied magnetic fields along the
crystallographic $b$ axis in the orthorhombic structure. The resultanting
volume magnetostriction of $\Delta V/V \approx-5.9\times 10^{-4}$ gives the
initial pressure dependence of $H_{\rm m}$ by employing the
Clausius-Clapeyron's equation, which agrees with previous pressure experiments.
Further, significant anisotropic magnetostriction (AMS), derived by subtracting
the averaged linear magnetostriction, was revealed. Contrary to the weakly
field-dependent AMS along the $a$ axis, those along the $b$ and $c$ axes show
strong field dependences with a similar magnitude but with opposite signs,
indicating its lattice instability. The relationship between characteristic
energy scales of magnetic fields and temperatures was discussed in terms of the
Gr\"uneisen parameters compared to the other $f$-electron systems. The volume
shrinkage in UTe$_2$ at $H_{\rm m}$, contrary to the volume expansion in
typical heavy fermion metamagnets, pushes to invoke the link with the valence
instability related to the itinerant-localized dual nature of the U magnetism.
