In general, Cech H^2 merely injects into derived H^2, so maybe one needs to be careful to distinguish these notions when formulating the question. (This sort of issue comes up for other coefficients when trying to relate cohomological Brauer group to the variant defined using Azumaya algebras, the latter being the Cech viewpoint.) I would imagine that stuff not coming from Cech H^2 is a bit harder to