= Solution
The station's proper rest mass is
$$
M_{\rm st}=4\pi r_{\rm st}^2\rho.
$$
A static local energy $dE_{\rm loc}$ at radius $r_{\rm st}$ contributes the redshifted energy
$$
dE_\infty=\sqrt{1-\frac{2M}{r_{\rm st}}}\,dE_{\rm loc}
$$
to the mass measured at infinity. To first order in $M_{\rm st}/M$, the exterior mass parameter is therefore
$$
\boxed{
M_{\rm out}
=M+4\pi r_{\rm st}^2\rho
\sqrt{1-\frac{2M}{r_{\rm st}}}
}.
$$
Terms quadratic in $\rho$ describe the shell's self-gravity and are neglected.
The exterior of the shell is vacuum and spherically symmetric. The <Birkhoff theorem> therefore makes it a <Schwarzschild spacetime> with mass $M_{\rm out}$:
$$
\boxed{
ds^2
=-\left(1-\frac{2M_{\rm out}}r\right)dt^2
+\left(1-\frac{2M_{\rm out}}r\right)^{-1}dr^2
+r^2d\Omega^2,
\qquad r>r_{\rm st}
}.
$$
This is the <redshifted gravitational mass of a static thin shell>.
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