Bistable dome shell structures exhibit snap-through instabilities that enable rapid shape changes and mechanical switching behavior. These elements have been reported as key components in soft valves, oscillators, and grippers, where they provide energy-efficient actuation and the ability to maintain stable configurations without continuous power input. However, their design involves facing nonlinear mechanics and high sensitivity to geometric parameters such as shell thickness, curvature, and height–radius ratio. Traditional design approaches rely on computationally expensive parametric sweeps or trial-and-error experimentation. While differentiable simulation has been successfully applied to beam-based bistable structures, its application to dome shell geometries remains less established. Differentiable simulations allow computing gradients of design objectives with respect to geometric and material parameters, enabling gradient-based optimization that can significantly accelerate the design process. In this research, an axisymmetric dome shell model under quasi-static loading conditions is used to study the existence of bistability, snap-through threshold, and force–displacement response using differentiable simulations. Key geometric parameters governing bistable behavior are identified, and their influence on snap-through characteristics is analyzed through gradient information. The results are validated against conventional finite element approaches to assess numerical accuracy and gradient reliability. The expected outcome is a set of guidelines on when differentiable simulation provides reliable gradients for bistable dome shell design, and what modeling choices are necessary to handle snap-through behavior in a stable and reproducible manner.