= Solution
The number flux through the surface is
$$
J_z=\frac{g_s}{(2\pi\hbar)^3}
\int_{p_z>p_0}\frac{p_z}{m}
\frac{d^3p}{e^{\beta(p^2/2m-E_F)}+1},
\qquad p_0=\sqrt{2m(E_F+V_0)}.
$$
Throughout this tail the occupation is Boltzmann to leading order. The two transverse Gaussian <integrals> contribute $2\pi m/\beta$, while
$$
\int_{p_0}^\infty\frac{p_z}{m}
e^{-\beta(p_z^2/2m-E_F)}dp_z
=\frac1\beta e^{-\beta V_0}.
$$
Therefore
$$
\boxed{J_z=\frac{g_sm}{4\pi^2\hbar^3}(k_BT)^2e^{-V_0/(k_BT)}.}
$$
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