No. Let $R'$ be any non-excellent dvr whatsoever, hence of equicharacteristic $p > 0$, and let $t \in R'$ be a uniformizer. Let $R = \mathbf{F}_p[t]_{(t)}$. The local inclusion $R \hookrightarrow R'$ has induced residue field extension that is separable since $\mathbf{F}_p$ is perfect. And of course $R$ is excellent for any of a million reasons. So this seems to be a counterexample.