  Doping MoS$_2$ with Re is known to alter the electronic, structural, and
tribological properties. Re-doped MoS$_2$ has been previously mainly studied in
monolayer or few-layer form, but can also be relevant for applications in
many-layer or bulk form. In this work, we use density functional theory to
explore the structure, phase stability, and Raman spectrum of bulk Re-doped
MoS$_2$. We consider the possibility of the Re dopant existing at different
locations and provide experimentally distinguishable characteristics of the
most likely sites: Mo-substitution and tetrahedral (t-) intercalation. We
demonstrate and benchmark a general approach to calculate Raman spectra of
doped materials with metallic densities of states by using atomic Raman tensors
from the pristine material. Applying this method to the metallic Re-doped
structures, we find characteristic shifts in the Raman-active peaks depending
on Re dopant position: redshifts in both A$_{\rm 1g}$ and E$_{\rm 2g}^1$ peaks
in the t-intercalated case versus a redshift for A$_{\rm 1g}$ and blueshift
(sometimes accompanied by a smaller redshifted peak) for E$_{\rm 2g}^1$ peaks
in the Mo-substituted case, which can be used to identify the dopant sites in
experimental samples. We analyze the interactions giving rise to these shifts.
