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 this, see https://mathoverflow.net/questions/354692/the-yoneda-pairing-hypercohomology-and-cup-product/354752#354752

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.
This is the classical bar construction,
in this case it is applied to the adjunction
between presheaves of chain complexes on C and Ob(C)-indexed abelian groups.