Show that language is context-free
Let $Q$ be the set of states in a DFA $D$ for $R$. We replace $\Gamma$ with $\Gamma \times Q$, and update the PDA so that the stack contains, apart from the original stack symbols, also the state that $D$ is in after reading the current word on the stack. The PDA only allows transitions if the element at the top of the stack corresponds to an accepting state of $D$, and also checks this condition at the very end.