= Solution
The Maxwell quadratic action has a null longitudinal direction: its momentum-space kinetic operator annihilates $k_\mu$, reflecting <gauge invariance> under addition of a gradient to the potential. It cannot be inverted on all four vector components before <gauge fixing>.
Add the <covariant gauge> density $\mathcal L_{\rm gf}=-(\partial_\mu A^\mu)^2/(2\xi)$. Away from the pole, introduce the <transverse and longitudinal momentum projectors>
$$
P^{\rm L}_{\mu\nu}=\frac{k_\mu k_\nu}{k^2},\qquad
P^{\rm T}_{\mu\nu}=\eta_{\mu\nu}-P^{\rm L}_{\mu\nu}.
$$
With one index raised they obey $P^{\rm T}+P^{\rm L}=I$, $(P^{\rm T})^2=P^{\rm T}$, $(P^{\rm L})^2=P^{\rm L}$ and $P^{\rm T}P^{\rm L}=0$. The gauge-fixed kinetic operator is $K=-k^2(P^{\rm T}+\xi^{-1}P^{\rm L})$. Its inverse times $i$ gives the <photon propagator>
$$
\boxed{D^{(\xi)}_{\mu\nu}(k)=\frac{-i}{k^2+i0}\left(\eta_{\mu\nu}-(1-\xi)\frac{k_\mu k_\nu}{k^2}\right)}.
$$
The pole factors are interpreted with the Feynman boundary prescription; the algebraic projector inversion is performed off the pole. The choice $\xi=1$ is <Feynman gauge>.
Changing $\xi$ changes only a longitudinal term proportional to $k_\mu k_\nu$. Between conserved currents it vanishes because $k_\mu J^\mu=0$. In full QED amplitudes, the <Ward-Takahashi identity> supplies the corresponding cancellations when all diagrams at an order are combined. Thus physical on-shell scattering is gauge-parameter independent, whereas off-shell Green functions and the propagator itself can depend on the gauge choice. For the charged scalar, derivative vertices and contact terms must likewise be included for this cancellation.
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