Multistable composite laminates represent a key technology for lightweight morphing structures, as they can undergo large and reversible shape changes. Their mechanical response can be tailored by bonding additional components such as stiffeners; however, this design strategy also introduces new failure mechanisms. In particular, stiffener integration may result in a loss of laminate bistability and initiate stiffener debonding due to crack propagation along the adhesive interface. This work examines the snap-through and post-snap behavior of bistable laminates with adhesively bonded stiffeners. A combined numerical approach based on nonlinear finite element analysis and linear elastic fracture mechanics, using the virtual crack closure technique (VCCT), is employed. Parametric studies are conducted to evaluate the influence of key geometric and material parameters on the interfacial energy release rate and the resulting post-snap configurations. Three mechanically relevant post-snap states are identified: bistable laminates with stiffener debonding, bistable laminates without stiffener debonding, and complete loss of bistability. These regimes are governed by the interaction between the stiffener-to-laminate bending stiffness ratio, the thermal expansion mismatch within the laminate, and the crack length at the laminate–stiffener interface. Configuration maps are introduced to delineate the parameter ranges associated with each regime. Experimental results successfully reproduce the numerical predictions.