This work addresses the multi-fidelity analysis-driven design of a thermal transport system based on the flow of liquid metal through a structural laminate as induced by a solid-state magneto-hydro-dynamic (MHD) pump. A full three-dimensional model of the thermal transport system is both simplified to a reduced-order algebraic model, which correctly captures trends in the global system response, and alternatively implemented in an finite element framework, which captures essential global and local aspects of the system response not attainable via reduced-order modeling. The predictions of each model are validated against previously published experimental data. It is shown in detail for the first time in the context of MHD systems that a multi-fidelity approach to the multi-objective design optimization problem can leverage both the speed of the algebraic model and the accuracy of the finite element model, leading to effective predictions of optimal system designs in a reasonable amount of time. A relatively new algorithm for multi-objective and parameterized Pareto optimization is employed, and a clear path of continued development is identified.