We study a resistance-distance invariant for Kekuléan molecular graphs obtained by assigning to each edge its Pauling bond order as an electrical conductance. The associated Pauling-weighted electrical network has finite pairwise effective resistances exactly when the positive bond-order support is connected, and in this case the corresponding resonance-weighted Kirchhoff index is well defined. The construction is a specialization of the weighted Kirchhoff index, but the weights are not external parameters: they are the edge frequencies in the Kekulé ensemble. Equivalently, the weighted Laplacian is the average of the Laplacians of all Kekulé structures. We prove trace normalization, a connected-support criterion, comparison inequalities, a conductance sensitivity formula, and the exact scaling law in the uniform bond-order case. In particular, uniform Pauling bond order recovers the ordinary Kirchhoff index only when the common order is one. Exact benchmark computations and bond-order profiles are given for benzene, naphthalene, anthracene, and phenanthrene. For the full linear-acene family we derive an exact finite-sum formula and the leading cubic growth of the resonance-weighted Kirchhoff index; these results provide reference values for a carefully stated catacondensed-benzenoid extremal problem.