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Here's a suggestion: take a geometric point $\bar{s} \rightarrow s$ above $s$. Then the stalk $\left(R^1 j_* P_{\eta}\right)_{\bar{s}}$ is computed as the Galois cohomology $H^1(K^{sh},P_{\eta})$ where $K^{sh}$ is the fraction field of the strict henselization of $R$. This follows from the discussion here, and the fact that the local ring $\mathcal{O}_{S,s}$ is isomorphic to $R$. The absolute Galois group of $K^{sh}$ is isomorphic to the inertia subgroup of the absolute Galois group of $K$, and since $P_{\eta} \rightarrow \eta$ extends to a smooth scheme over $S$, general base change theorems tell you that this implies that inertia acts trivially. This allows you to describe the group $H^0(S,R^1j_*P_{\eta})$ more explicitly.