The last few decades have led to the rise of research focused on propulsion and control systems for bio-inspired unmanned underwater vehicles (UUVs), which provide more maneuverable alternatives to traditional UUVs in underwater missions. Recent work has explored the use of time-series neural network surrogate models to predict thrust and power from vehicle design and fin kinematics. We develop a search-based inverse model that leverages kinematics-to-thrust and kinematics-to-power neural network models for control system design. Our inverse model finds a set of fin kinematics with the multi-objective goal of reaching a target thrust under power constraints while creating a smooth kinematics transition between flapping cycles. We demonstrate how a control system integrating this inverse model can make online, cycle-to-cycle adjustments to prioritize different system objectives, with improvements in increasing thrust generation or reducing power consumption of any given movement upwards of 0.5 N and 3.0 W in a range of 2.2 N and 9.0 W. As propulsive efficiency is of utmost importance for flapping-fin UUVs in order to extend their range and endurance for essential operations but lacks prior research, we develop a non-dimensional figure of merit (FOM), derived from measures of propulsive efficiency, that is able to evaluate different fin designs and kinematics, and allow for comparison with other bio-inspired platforms. We use the developed FOM to analyze optimal gaits and compare the performance between different fin materials, providing a better understanding of how fin materials affect thrust generation and propulsive efficiency and allowing us to inform control systems and weight for efficiency on the developed inverse gait-selector model.