The pure-trace source calculation

The coefficient $\mathfrak{d}$ includes $\lambda$. The coefficient ${\mathfrak{d}}_1$ does not. Likewise, $\mathsf{q}$ is not multiplied by an additional frequency factor inside its definition.

At fixed $t$, after removing the common exponential, (eq:JDeval) is a polynomial in $\lambda$. Its degree-two coefficient is $\sigma (t)\mathsf{q}$. Where that coefficient is nonzero, a degree-one correction cannot cancel it as a polynomial family. This is a source statement, not a conclusion about a retarded response or complete insertion.

At $\lambda =0$, the displayed source vanishes. For $\lambda >0$ and $t\ge \epsilon$,

Thus the source is not in ${C_c}^{\infty }({M_T}^{\circ })$: it is nonzero at every spatial point at such a slab time.