I think you want your functor to be a delta-functor from the abelian category of representations to D^b_{Coh}(R) which is moreover compatible with tensor products. By the way. I suggest we replace D^b_{Coh}(R) by the derived category of perfect complexes, just to be in vogue.

A special case is where R = k. Then D^b_{Coh}(k) is the category of finite dimensional graded k-vector spaces endowed with a suitable structure of derived category. Suppose we only look at those functors where delta is always zero. (By the way, if the group scheme is linearly reductive, this will always be the case.) Then we obtain simply a k-linear faithful exact tensor functor into the category of graded vector spaces. If we forget the grading, then we obtain a G-torsor over k as you say. Suppose that this torsor is trivial, e.g., if k is algebraically closed. Then what's left over is a grading on each representation of G, compatible with tensor product. Such a thing is given by a cocharacter, i.e., a group scheme homomorphism G_m ---> G.

Thus, in the very special case just discussed, there seems to be a little bit more structure, than for usual tensor functors. Before trying to answer your very interesting question in full, we should try to answer it in the case where G is the additive group over k. My feeling would be that delta still has to be zero...