It seems we are miscommunicating. If you restrict the policies to simple rules, both should be doable. The hard part is doing full Ramsey manually as the optimal policy would have a coefficient in front of every state variable.
The cooperative case should be straightforward as you describe it: maximizing joint welfare. The hard thing is the Nash problem as now every country maximizes its own welfare taking the foreign policy as given. Here you are looking for the intersection of the best responses. As this is only a fixed point, you might be able to solve for such a fixed point. But in this case, you might need two loops that iteratively search for the fixed point. If we are talking about symmetric countries, this should considerably simplify because you are maximizing domestic welfare over the domestic policy rule subject to the other country running the same policy.