This paper presents a methodology for the design of deployable tube flexures made of ultra-thin carbon fiber composite. The proposed process aims to enhance viscoelastic resilience on recovery time after long-term stowage through the design of cut-out geometries. Viscoelasticity is modeled using an experimentally determined Prony series master curve. Bayesian optimization is employed to determine cut-out shapes based on the recovery time and Hashin failure criteria. Partial dependent plots are implemented to interpret the effect of cut-out formation on deployment performance. These reveal optimal profiles of cut-out shapes to achieve a desired operational performance. The efficiency of the proposed methodology is validated by experiment.