Thermal runaway during cure limits robust process design for thick composite laminates, especially when shell curvature and non-uniform heat transfer alter local heat removal. We present a semi-analytical framework for estimating safe curing limits in curved composite shells by reducing the three-dimensional thermo-kinetic problem to a locally one-dimensional through-thickness stability problem evaluated pointwise over the mid-surface. The resulting criterion is expressed in terms of a critical Damköhler number and separates geometry and boundary heat transfer, represented by a stability factor depending on principal curvatures and Biot numbers, from chemistry and processing temperature, represented by Arrhenius scaling and an effective kinetic factor. The geometry-dependent stability factor is obtained from a nonlinear boundary-value problem and represented by compact differentiable response surfaces for symmetric and asymmetric boundary conditions. Validation against fully coupled transient simulations confirms the predicted separation over the investigated parameter range. A complementary analytical and quasi-three-dimensional flux-ratio assessment shows that lateral heat transport remains small for the representative smooth shell geometries studied, with strongly anticlastic regions providing the most restrictive cases. The framework enables rapid curvature-based stability sweeps, identification of critical locations, estimation of safe thickness limits, and practical screening of cure-cycle modifications without full three-dimensional simulation.