= Solution
For a mixed tensor,
$$
\nabla_\mu T^\mu{}_{\nu}
=\partial_\mu T^\mu{}_{\nu}
+\Gamma^\mu{}_{\mu\lambda}T^\lambda{}_{\nu}
-\Gamma^\lambda{}_{\mu\nu}T^\mu{}_{\lambda}.
$$
Write $T=\bar T+\delta T$ and $\Gamma=\bar\Gamma+\delta\Gamma$. The homogeneous background already satisfies $\bar\nabla_\mu\bar T^\mu{}_{\nu}=0$, and discarding products of two perturbations leaves
$$
\boxed{
\delta(\nabla_\mu T^\mu{}_{\nu})
=\partial_\mu\delta T^\mu{}_{\nu}
+\bar\Gamma^\mu{}_{\mu\lambda}\delta T^\lambda{}_{\nu}
-\bar\Gamma^\lambda{}_{\mu\nu}\delta T^\mu{}_{\lambda}
+\delta\Gamma^\mu{}_{\mu\lambda}\bar T^\lambda{}_{\nu}
-\delta\Gamma^\lambda{}_{\mu\nu}\bar T^\mu{}_{\lambda}}.
$$
This is simply the first variation of the tensor's <covariant derivative>.
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