The $G((t))$ action and the $Rep(G^\vee)$ [or equivalently of $G^\vee$ itself after deequivariantization, as Victor explains] are of quite different natures -- the former is a "smooth" action, and the latter an "algebraic" or "analytic" actions (the adjectives smooth and analytic come from analogy with p-adic rep theory).
i.e. there are many kinds of notion of group action, and they are (to me) most conveniently summarized by describing the corresponding notion of group algebra which acts. An algebraic action of a group on a category is an action of the "quasicoherent group algebra" of G, ie the monoidal category of quasicoherent sheaves wrt convolution. (though I'd feel much safer if we said all this in a derived context, makes me uneasy otherwise).
A smooth action is an action of the monoidal category of D-modules on G, the "smooth group algebra" -- analog of smooth functions on a p-adic group. Such an action is the same as an algebraic action, which is infinitesimally trivialized. Such examples are studied in Chapter 7 of Beilinson-Drinfeld's Hecke manuscript and the appendix to the long paper by Gaitsgory-Frenkel, in particular.

PS the "equivariantization" dictionary between categories over BH and categories with H action is a nice simple case of descent --- you describe things over BH as things over a point with descent data, that descent data is given by the map H --> pt,
the two maps H x H---> H, and so on. When you assemble this together (most efficiently using the Barr-Beck theorem) you get the desired dictionary. (Of course if you want to consider categories as forming a 2-category you'd need a 2-categorical version of Barr-Beck, but for most practical purposes I know of you can get by with the current
Lurie [$(\infty,$]1-categorical version.