A current challenge for the Advanced Propulsion Experiment (APEX) CubeSat, under development by Missouri University of Science and Technology, is the accurate determination of thrust for its experimental multimodal propulsion system main payload. A consider batch least-squares filter is proposed that processes downlinked dual-frequency GPS pseudorange data to estimate the average thrust magnitude of a maneuver in conjunction with the initial dynamic states, static coefficients of drag and solar radiation pressure, and GPS receiver clock biases from each measurement epoch. Two Monte Carlo analyses were performed to test the statistical consistency of the filter and the robustness under the influence of unmodeled dynamics. The filter was found to perform nominally with matching truth and filter dynamics models and acceptably when the truth model used to synthesize measurements was a higher fidelity than the filter model. Filter performance was assessed using flight data from NASA's Gravity Recovery and Climate Experiment Follow-On mission. A trade study was conducted to identify acceptable orbital regimes for the APEX mission. Thrust estimates using this method were shown to meet APEX mission requirements for a range of acceptable orbits.