= Solution
{c}
In <unitary gauge> the quadratic <action> is that of a <Proca field>. Use <Minkowski metric> $(+---)$ and add a source $J_\mu A^\mu$. <Integration by parts>, dropping the boundary term, gives
$$
S[A]=\frac12\int d^4x\,A_\mu K^{\mu\nu}A_\nu,\qquad
K^{\mu\nu}=g^{\mu\nu}(\Box+M^2)-\partial^\mu\partial^\nu.
$$
With <Fourier transform> convention $A(x)=\int d^4k\,e^{-ik\cdot x}A(k)/(2\pi)^4$, its kernel is
$$
K^{\mu\nu}(k)=(M^2-k^2)g^{\mu\nu}+k^\mu k^\nu.
$$
For $k^2\ne0$, introduce the <transverse and longitudinal momentum projectors>
$$
P_T^{\mu\nu}=g^{\mu\nu}-\frac{k^\mu k^\nu}{k^2},\qquad
P_L^{\mu\nu}=\frac{k^\mu k^\nu}{k^2}.
$$
They obey $P_TP_L=0$ and $P_T+P_L=1$ as mixed-index <linear operators>. Hence
$$
K=(M^2-k^2)P_T+M^2P_L,\qquad
K^{-1}=\frac{P_T}{M^2-k^2}+\frac{P_L}{M^2}
=-\frac{g-kk/M^2}{k^2-M^2}.
$$
The final expression is regular at $k^2=0$; the apparent singularities of the individual <projectors> have cancelled.
The normalized <Gaussian functional integral> is evaluated by <completing the square> with $A\mapsto A+K^{-1}J$:
$$
\frac{Z[J]}{Z[0]}=
\frac{\int\mathcal DA\,\exp\{i(\frac12AKA+JA)\}}{\int\mathcal DA\,\exp\{i\frac12AKA\}}
=\exp\left[-\frac i2\int\frac{d^4k}{(2\pi)^4}J_\mu(-k)(K^{-1})^{\mu\nu}(k)J_\nu(k)\right].
$$
The source-independent <functional determinant> cancels in this ratio. Two source derivatives insert two fields, so the covariant <Feynman propagator> is $iK^{-1}$, with the <Feynman i-epsilon prescription> at the physical pole:
$$
\boxed{D_{\mu\nu}(k)=\frac{-i}{k^2-M^2+i0}\left(g_{\mu\nu}-\frac{k_\mu k_\nu}{M^2}\right),\qquad
D_{\mu\nu}(x-y)=\int\frac{d^4k}{(2\pi)^4}e^{-ik\cdot(x-y)}D_{\mu\nu}(k).}
$$
This is the <Proca propagator>. Its pole residue has the three physical <particle polarizations> of a massive spin-one <gauge boson>. No additional gauge-fixing term is needed to invert this already gauge-fixed massive quadratic <action>. In a full interacting <Higgs mechanism> calculation, any field-dependent <unitary gauge> measure belongs to the interacting theory; the free quadratic <gauge boson> <propagator> above is unaffected.
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