= Solution
The components of the <momentum operator> do not generally commute with a position-dependent <magnetic vector potential>. Expanding <minimal coupling> with the ordering intact gives
$$
H=\frac{p^2}{2m}+V+
\frac e{2m}\bigl(\mathbf p\cdot\mathbf A+\mathbf A\cdot\mathbf p\bigr)
+\frac{e^2}{2m}\mathbf A^2.
$$
Acting on a <wavefunction> shows that $\mathbf p\cdot\mathbf A-\mathbf A\cdot\mathbf p=-i\hbar\nabla\cdot\mathbf A$. In the <Coulomb gauge> the divergence vanishes, so the two cross terms agree. Discarding the quadratic term in the weak radiation amplitude gives
$$
\boxed{H=H_0+H_1(t)+O(A^2),\qquad
H_0=\frac{p^2}{2m}+V,\qquad
H_1(t)=\frac em\mathbf p\cdot\mathbf A(\mathbf r,t).}
$$
The equality of the two operator orderings follows from the gauge condition, not from treating $\mathbf p$ and $\mathbf A$ as commuting numbers component by component.
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