Cup products in sheaf (and presheaf) cohomology are often easy to compute by resolving the source (in the projective model structure, say), not the target. For an example of resolving the source in this manner, see The Yoneda pairing, hypercohomology, and cup product
In the case under consideration, one can equip the category of presheaves of chain complexes on C with a projective model structure. The latter has an explicit cofibrant replacement functor, which can be used to write down an explicit projective resolution. The cofibrant replacement functor is precisely the classical bar construction applied to the adjunction between presheaves of chain complexes on C and Ob(C)-indexed chain complexes.