Correcting gradual position drift is a challenge in long-term subsea navigation. Though highly accurate, modern inertial navigation system (INS) estimates will drift over time due to the accumulated effects of sensor noise and biases, even with acoustic aiding from a Doppler velocity log (DVL). The raw sensor measurements and estimation algorithms used by the DVL-aided INS are often proprietary, which restricts the fusion of additional sensors that could bound navigation drift over time. In this letter, the raw sensor measurements and their respective covariances are estimated from the DVL-aided INS output using semidefinite programming tools. The estimated measurements are then augmented with laser-based loop-closure measurements in a batch state estimation framework to correct planar position errors. The heading uncertainty from the DVL-aided INS is also considered in the estimation of the updated positions. The pipeline is tested in simulation and on experimental field data. The proposed methodology reduces the long-term navigation drift by more than 30 times compared to the DVL-aided INS estimate.