Single-qubit sensing protocols can be used to measure qubit-bath coupling parameters. However, for sufficiently large coupling, the sensing protocol itself perturbs the bath, which is predicted to result in a characteristic response in the sensing measurements. Here, we observe this bath perturbation, also known as a quantum quench, by preparing the nuclear spin bath of a nitrogen-vacancy (NV) center in polarized initial states and performing phase-resolved spin-echo measurements on the NV electron spin. These measurements reveal a time-dependent phase determined by the initial state of the bath. We derive the relationship between the sensor phase and the Gaussian spin-bath polarization and apply it to reconstruct both the axial and transverse polarization components. Using this insight, we optimize the transfer efficiency of our dynamic nuclear polarization sequence. This technique for directly measuring bath polarization may assist in preparing high-fidelity quantum memory states, improving nanoscale NMR methods, and investigating non-Gaussian quantum baths.