= Solution
The host mass inside $D$ follows from the same <singular isothermal sphere>:
$$
M_h(<D)=\frac{2\sigma_h^2D}{G}=\frac{2(3\times10^5)^2(1.5\times10^{21})}{7\times10^{-11}}\,\mathrm{kg}=1.93\times10^{12}M_\odot.
$$
Using the supplied host luminosity gives
$$
\boxed{\Upsilon_h(<50\,\mathrm{kpc})\simeq20M_\odot/L_\odot.}
$$
This treats essentially all the stated optical light as contained in that aperture; a measured luminosity profile would refine that convention. Keeping consistent unrounded quantities,
$$
\frac{\Upsilon_d}{\Upsilon_h}=\frac{m_d}{M_h}\frac{L_h}{L_d}=\frac1{\sqrt2}\left(\frac{\sigma_d}{\sigma_h}\right)^3\left(\frac{\sigma_h}{\sigma_d}\right)^4=\frac{\sigma_h}{\sqrt2\sigma_d}\simeq21.
$$
Thus \b[the dwarf ratio is about twenty times the parent ratio]; the rough equal-density tidal convention would give a factor thirty.
There is no logical contradiction in two objects sharing $L\propto\sigma^4$ but having different <mass-to-light ratios>. A <virial theorem> scaling gives $M\propto\sigma^2R$, so that $M/L\propto R/\sigma^2$. Equal ratios additionally require homologous relevant radii scaling as $R\propto\sigma^2$. Here the satellite's <tidal radius> scales as $\sigma_d$ in the fixed host field, not as $\sigma_d^2$. Equating the two ratios would require $r_t=D(\sigma_d/\sigma_h)^2\simeq56\,\mathrm{pc}$, far smaller than the derived tidal radius.
The heuristic derivation of the <Faber-Jackson relation> with common <mass-to-light ratio> and common <surface brightness> made auxiliary structural assumptions; the empirical relation alone cannot enforce them. Furthermore, the <aperture dependence of mass-to-light ratio> matters: a satellite's whole tidal mass and a host's $50\,\mathrm{kpc}$ halo mass are not homologous half-light-aperture masses. <Dark matter> need not follow stellar light, and a <dwarf spheroidal galaxy> is not automatically a scaled giant <elliptical galaxy>. The model calculates total gravitating mass: reading its density literally as stars alone would imply an implausibly large purely stellar ratio, exposing the limits of that interpretation.
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