The success of model-based multifunctional material design efforts relies on the proper development of multiphysical models and advanced optimization algorithms. This paper addresses both in the context of a structure that includes a liquid metal (LM) circuit for integrated cooling. We demonstrate for the first time on a complex engineering problem the use of a parameterized approach to design optimization that solves a family of optimization problems as a function of parameters exogenous to the subsystem of interest. This results in general knowledge about the capabilities of the subsystem rather than a restrictive point solution. We solve this specialized problem using the predictive parameterized Pareto genetic algorithm (P3GA) and show that it efficiently produces results that are accurate and useful for design exploration and reasoning. A population seeding approach allows an efficient multifidelity approach that combines a computationally efficient reduced-fidelity algebraic model with a computationally intensive finite-element model. Using data output from P3GA, we explore different design scenarios for the LM thermal management concept and demonstrate how engineers can make a final design selection once the exogenous parameters are resolved.