Your target BⁿA is an HZ-module. HZ-modules are Quillen equivalent to chain complexes, and the delooping functor on HZ-modules corresponds to the shift functor on chain complexes.

The space C(H_i,BⁿA) can be computed as BⁿC(H_i,A) (this is how I interpret the rather vague statement “BⁿA is fibrant on its simplices”), and by the above correspondence this is simply a shift of C(H_i,A), i.e., a shift of your cosimplicial object C(H_∙,A).

Cosimplicial homotopy limits of chain complexes can be computed as the totalization (see Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes) 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_−n will give you Hⁿ(X,A).

This point of view is explained in Ken Brown's famous paper.

Your target BⁿA is an HZ-module. HZ-modules are Quillen equivalent to chain complexes, and the delooping functor on HZ-modules corresponds to the shift functor on chain complexes.

The space C(H_i,BⁿA) can be computed as BⁿC(H_i,A) (this is how I interpret the rather vague statement “BⁿA is fibrant on its simplices”), and by the above correspondence this is simply a shift of C(H_i,A), i.e., a shift of your cosimplicial object C(H_∙,A).

Cosimplicial homotopy limits of chain complexes can be computed as the totalization (see Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes) 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_−n will give you Hⁿ(X,A).

This point of view is explained in Ken Brown's famous paper.