= Solution
The local <U(1) gauge symmetry> is
$$
\psi\mapsto e^{ie\chi(x)}\psi,
\qquad
\bar\psi\mapsto\bar\psi e^{-ie\chi(x)},
\qquad
A_\mu\mapsto A_\mu-\partial_\mu\chi.
$$
The global phase subgroup has <Noether current> and charge
$$
j^\mu=\bar\psi\gamma^\mu\psi,
\qquad
Q=\int d^3x\,j^0=\int d^3x\,\psi^\dagger\psi.
$$
Multiplying the fermion equation by $\bar\psi$, its adjoint by $\psi$, and subtracting shows $\partial_\mu j^\mu=0$ on shell. Therefore $\dot Q=-\int d^3x\,\boldsymbol\nabla\cdot\mathbf j=0$ under vanishing boundary flux.
The source in Maxwell's equation is
$$
J^\nu=e j^\nu-2i\lambda\partial_\mu
(\bar\psi[\gamma^\mu,\gamma^\nu]\psi).
$$
The second term is an identically conserved <magnetization current>, because it is the divergence of an antisymmetric tensor. It changes the local source but contributes only a boundary term to the total charge.
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