Proof Idea: How to prove the intersection of regular language and CFL is a CFL?
Look at the product construction for DFAs, it will give you a hint of what the state space of a PDA accepting the intersection will look like.
You can convert $D$ to a PDA, but you have more information than that. It doesn't need to touch the stack at all, since it can be accepted by a DFA.


These should get you close to the answer.