Separable quotients for less-than-barrelled function spaces

We solve the separable quotient problem for spaces Cp(X) and Cc(X) that are dual locally complete (dlc), greatly extending the positive solution for barrelled Cc(X). Indeed, ours is either the best or next-to-best weak barrelledness extension possible. Yet some interesting non-dlc spaces Cp(X) and Cc(X) also admit separable quotients.