Your target B<sup>n</sup>A is an H**Z**-module.
H**Z**-modules are Quillen equivalent to chain complexes,
and the delooping functor on H**Z**-modules corresponds to the shift functor on chain complexes.

The space C(H<sub>i</sub>,B<sup>n</sup>A) can be computed as B<sup>n</sup>C(H<sub>i</sub>,A) (this is how I interpret the
rather vague statement “B<sup>n</sup>A is fibrant on its simplices”),
and by the above correspondence
this is simply a shift of C(H<sub>i</sub>,A),
i.e., a shift of your cosimplicial object C(H<sub>∙</sub>,A).

Cosimplicial homotopy limits of chain complexes
can be computed as the totalization (see https://mathoverflow.net/questions/194010/reference-for-homotopy-colimits-of-cochain-complexes-via-totalization-of-dou) of the corresponding bicomplex obtained via Dold-Kan,
which shows that your construction is equivalent to nLab's.

> In derived functor cohomology one takes cochain resolutions of A, which again by co-Dold-Kan is the same thing as cosimplicial resolutions. Can one also do this the nonabelian setting?

Yes, injective resolutions of sheaves can be expressed in the nonabelian setting as fibrant replacements in the injective model structure on simplicial presheaves.
This would give a different but equivalent computation
to the above one (which uses the projective model structure).

> Is there some way to make sense of this, i.e. perhaps by turning C(X,A)C(X,A) into some kind of truncated spectrum?

Yes, start with the original sheaf of abelian groups,
apply the Eilenberg-MacLane spectrum functor (respectively simply
place it in chain degree 0), and then ∞-sheafify the resulting presheaf
of spectra (respectively chain complexes).
Evaluating the resulting ∞-sheaf on some X and taking π<sub>−n</sub>
will give you H<sup>n</sup>(X,A).

This point of view is explained in [Ken Brown's famous paper](https://ncatlab.org/nlab/show/BrownAHT).