Solution
= Solution
Use the spatial <transverse-longitudinal decomposition>
$$
A_0=\phi,
\qquad
A_i=A_i^V+\partial_i\chi,
\qquad
\partial_iA_i^V=0.
$$
Under the <U(1) gauge symmetry> $A_\mu\mapsto A_\mu+\partial_\mu\epsilon$,
$$
\phi\mapsto\phi+\dot\epsilon,
\qquad
\chi\mapsto\chi+\epsilon,
\qquad
A_i^V\mapsto A_i^V.
$$
The assumption $\nabla^2\epsilon\ne0$ makes the longitudinal split unambiguous. The scalar
$$
\boxed{\Phi_A=\phi-\dot\chi}
$$
is therefore <gauge-invariant>.