What's going on here formally in the category case is that the inclusion $A \subset B$ extends to a map $f: A \to [B^{\operatorname{op}}, \mathcal{V}]$ (where $\mathcal{V}$ is the category you're enriching over) via the Yoneda embedding. The Yoneda embedding is the free cocompletion of a category (i.e., the completion under weighted colimits), so this map extends to a cocontinuous functor $\widehat{f}: [A^{\operatorname{op}}, \mathcal{V}] \to [B^{\operatorname{op}}, \mathcal{V}]$, given by the weighted colimit formula you indicated. This functor is right adjoint to $\operatorname{Hom}(f-, -): [B^{\operatorname{op}}, \mathcal{V}] \to [A^{\operatorname{op}}, \mathcal{V}]$, which is just the restriction functor in this case. (What I've said works more generally for any such $f$, not just one induced by an inclusion of categories.)

Everything I have said should extend more or less identically to the setting of bicategories. A pseudofunctor between bicategories gives you an induction pseudofunctor between the corresponding presheaf categories by a weighted bicolimit formula analogous to the weighted colimit formula in the 1-categorical setting. This should be right biadjoint to the obvious restriction pseudofunctor, although I haven't checked the details (and I don't know if this is written up anywhere).

The situation you are interested in seems a bit more delicate, because you want to work in the setting where your hom-categories are enriched. To formulate all of this properly, one would need a notion of "weak enrichment," which generalizes the theory of bicategories. I've heard that there are people working on this, but I don't know of any available sources yet.

Note that while in principle every "weakly $\mathcal{V}$-$\operatorname{Cat}$-enriched category" may be equivalent to a category enriched in the underlying 1-category $\mathcal{V}$-$\operatorname{Cat}$ (and indeed, you may only be interested in the latter kinds of categories), the $\mathcal{V}$-$\operatorname{Cat}$-enriched setting probably isn't flexible to properly handle the kind of "representations" you would like to handle.

**Edit**: I really just wanted to fix the spelling of "principle" above, but as long as I'm bumping a two-year-old question, I might as well link to a recent paper by Garner and Shulman that develops the theory of bicategories enriched in a monoidal bicategory. In particular, they show that the bicategory of (certain) modules over an enriched bicategory is the free cocompletion under (certain) weighted colimits, which allows you to describe induction as a weighted colimit as in the 1-categorical setting.