The global demand for grid-scale energy storage continues to increase. Carnot batteries (CBs) are not geographically constrained and consist of mature components. Furthermore, charging power, discharging power, and storage capacity can be sized independently and tailored to the intended use case. Designing an optimal CB also requires considering the resulting operational behavior. While design and operation are frequently addressed separately, only simultaneous consideration can ensure an overall optimum. To this end, we extend our hybrid mechanistic/data-driven reduced-space CB model, used for working fluid (WF) screening in Lüthje et al. (2026) [Optim Eng, 27, 587–616], to include component sizing, cost correlations, and optimal operation for price arbitrage in the German day-ahead market. To reduce computational complexity, we solve both heat transfer calculations and the optimal operation problem offline, and embed surrogate models, respectively predicting heat transfer coefficients and revenue as functions of the battery parameters. We compute the Pareto curve between capital cost minimization and revenue maximization for a CB with a nominal charging power of 50 MW using MAiNGO v0.10.2, applying reliability branching to improve performance. Throughout, we consider deterministic global optimization, noting that by definition, the Pareto curve considers global solutions. Results vary strongly with the considered electricity price profile and WF. A CB, without restrictions on the choice of WF, would have been profitable in the German day-ahead market in 2022 and 2024. However, this is not the case when only non-toxic, environmentally friendly WFs are considered, emphasizing the importance of research on novel WFs.