I would like to know if there exists a formulation/incarnation of the Fourier-Mukai duality in terms of the corresponding Frobenius manifold constructed by Barannikov and Kontsevich: http://arxiv.org/pdf/alg-geom/9710032
Yes, a derived equivalence will give an equivalence of the corresponding Frobenius manifolds. First a derived equivalence induces an isomorphism of deformation spaces of the two categories: the deformation theory of the derived category is controlled by the Hochschild cochains with its differential graded Lie algebra structure, and this can be recovered intrinsically from the derived category (or better its enhanced versions) and so is preserved under a Fourier-Mukai transform. Next if the category is Calabi-Yau then it intrinsically defines (following Costello, Kontsevich-Soibelman and later Lurie) an extended 2d topological field theory (aka TCFT), and so again a Fourier-Mukai transform induces an isomorphism of TFTs. Finally to get a Frobenius manifold structure you want a topological string theory, in other words you want to extend the field theory to the boundary of Deligne-Mumford space. But again (following Konstevich et al) this amounts to the derived category satisfying the degeneration conjecture (roughly the circle action on Hochschild homology should be "trivial"), but this is known in the geometric setting (the Hodge theorem) and in any case again this is an intrinsic property of the derived category, so preserved by derived equivalence. The structure of (germ of) Frobenius manifold is describing the genus zero part of the Deligne-Mumford action on the Hochschild cohomology (see papers of Getzler identifying the formal Frobenius operad with those of "gravity", aka compactified genus zero moduli spaces).
Anyway in short, the derived category (or better an enhanced, dg or $A_\infty$) version controls all the other aspects of the B-model TFT, so all the other constructions in mirror symmetry on the B-side can be recovered intrinsically from it (and are therefore invariant under derived equivalence).