We calculate the frequency-dependent admittance of a phase-biased Josephson junction spanning a magnetic impurity or a spinful Coulomb-blockaded quantum dot. The local magnetic moment gives rise to Yu-Shiba-Rusinov bound states, which govern the subgap absorption as well as the inductive response. We model the system by a superconducting spin-polarized exchange-cotunnel junction and calculate the linear current response to an ac bias voltage, including its dependence on phase bias as well as particle-hole and source-drain coupling asymmetry. The corresponding inductive admittance is analyzed and compared to results of a zero bandwidth, as well as an infinite-gap approximation to the superconducting Anderson model. All three approaches capture the interaction-induced 0-π transition, which is reflected as a discontinuity in the adiabatic inductive response.