This article explores the optimization of plant characteristics and controller parameters for electrified mobility. Electrification of mobile transportation systems, such as automobiles and aircraft, presents the ability to improve key performance metrics such as efficiency and cost. However, the strong bidirectional coupling between electrical and thermal dynamics within new components creates integration challenges, increasing component degradation, and reducing performance. Diminishing these issues requires novel plant designs and control strategies. The electrified mobility literature provides prior studies on plant and controller optimization, known as control co-design (CCD). A void within these studies is the lack of model predictive control (MPC), recognized to manage multi-domain dynamics for electrified systems, within CCD frameworks. This article addresses this through three contributions. First, a thermo-electromechanical hybrid electric vehicle (HEV) powertrain model is developed that is suitable for both plant optimization and MPC. Second, simultaneous plant and controller optimization is performed for this multi-domain system. Third, MPC is integrated within a CCD framework using the candidate HEV powertrain model. Results indicate that optimizing both the plant and MPC parameters simultaneously can reduce physical component sizes by over 60% and key performance metric errors by over 50%.