= Solution
Integrate <galactic stellar mass>, rather than galaxy count, against the <galaxy stellar mass function>. With $x=M_*/M_{*,\rm ch}$, the comoving <cosmic stellar mass density>, evaluated by the <stellar mass density from a square-root exponential cutoff>, is
$$
\rho_*^{\rm com}=\phi_{\rm ch}M_{*,\rm ch}\int_0^\infty x^{-3/4}e^{-\sqrt x}\,dx.
$$
Put $u=\sqrt x$, so $dx=2u\,du$ and $x^{-3/4}=u^{-3/2}$. The integral is $2\Gamma(1/2)=2\sqrt\pi$, giving
$$
\boxed{\rho_*^{\rm com}=2\sqrt\pi\,\phi_{\rm ch}M_{*,\rm ch}\simeq3.54\times10^8\,M_\odot\,\mathrm{Mpc^{-3}}.}
$$
The supplied present mean matter density and $\Omega_m=0.25$ imply $\rho_{\rm crit,0}=3.5\times10^{10}/0.25=1.4\times10^{11}\,M_\odot\,\mathrm{Mpc^{-3}}$. Therefore
$$
\boxed{\frac{\rho_*^{\rm com}}{\rho_{\rm crit,0}}\simeq2.53\times10^{-3},\qquad \frac{\rho_*^{\rm com}}{\bar\rho_{m,0}}\simeq1.01\times10^{-2}.}
$$
This is about \b[$0.25\%$ of today's <critical density>], or \b[$1\%$ of today's mean matter density], expressed per present <comoving volume>. The proper stellar density at $z=3$ is $64\rho_*^{\rm com}$; if comparing that proper density directly with today's <critical density> its ratio is $0.162$. It must not be confused with the usual comoving-density comparison. The low-mass number integral diverges if extrapolated to zero mass, but the mass integral converges; real populations require a low-mass cutoff.
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