= Solution
Under $a^I_\alpha\mapsto a^I_\alpha+\partial_\alpha\Lambda^I$, the Chern--Simons term changes by
$$
\frac1{4\pi\hbar}K_{IJ}
\epsilon^{\alpha\beta\gamma}
\partial_\alpha\Lambda^I\partial_\beta a^J_\gamma
=\partial_\alpha\left[
\frac1{4\pi\hbar}K_{IJ}
\epsilon^{\alpha\beta\gamma}
\Lambda^I\partial_\beta a^J_\gamma
\right],
$$
because the contraction with two commuting derivatives vanishes. The source term changes by
$$
-j_I^\alpha\partial_\alpha\Lambda^I
=-\partial_\alpha(j_I^\alpha\Lambda^I)
+\Lambda^I\partial_\alpha j_I^\alpha.
$$
<Conservation law>[Current conservation] removes the last term, so
$$
\boxed{\delta\mathcal L=\text{a total derivative}}.
$$
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