But the general localization procedure works for choices of $V$ different from and more general than SSet with its standard model structure. In particular it should work for

For these cases the local model structure on $V$-valued presheaves should model, respectively, $(n,r)$-category valued sheaves/stacks and $(\infty,1)$-operad valued sheaves/stacks.

References

The general localization result is apparently due to

Clark Barwick, On left and right model categories and left and right Bousfield localization Homology, Homotopy and Applications, vol. 12(2), 2010, pp.245–320 (pdf)

which considers the ?ech cover?-localization assuming $V$ to be monoidal and

Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (pdf)

which apparently does the hypercoverdescent and without assuming $V$ to be monoidal.