= Solution
The free complex <Proca field> has <Poincaré symmetry>, with $B_\mu$ transforming as a spacetime <four-vector>, and the global internal <U(1)> <symmetry>
$$
B_\mu\longmapsto e^{i\alpha}B_\mu,\qquad B_\mu^*\longmapsto e^{-i\alpha}B_\mu^*.
$$
The free theory is also invariant under charge conjugation $B_\mu\leftrightarrow B_\mu^*$ and the usual parity and <time reversal> transformations of a vector field. The internal real-field description consists of two equal-mass real <Proca fields>, with an $O(2)$ symmetry containing the phase rotations and charge conjugation.
A gradient shift of $B_\mu$ leaves $G_{\mu\nu}$ unchanged, because partial derivatives commute. It changes the mass term by
$$
\Delta\mathcal L=M^2\left[-B_\mu^*\partial^\mu\lambda-(\partial_\mu\lambda^*)B^\mu+(\partial_\mu\lambda^*)\partial^\mu\lambda\right].
$$
This is generally nonzero: for example, $B_\mu=0$ and real $\lambda=at$ produce $\Delta\mathcal L=M^2a^2$. Thus a massive <Proca field> has no such <gauge symmetry>. The massless kinetic term alone would have it.
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