Recent advances in augmented Lagrangian methods for infinite-dimensional constrained optimization have significantly broadened this classical algorithmic framework. However, cutting-edge approaches like ALESQP were not designed to handle the nonsmooth composite objective functions typically found in risk-averse stochastic optimization models. Conversely, methods such as the primal-dual risk minimization (PD-Risk) algorithm, developed for minimizing structured nonsmooth risk measures, do not account for simulation-based equality constraints or complex geometric constraints. We remedy this gap by fusing PD-Risk with ALESQP to introduce a new method: RALES. RALES is a provably convergent algorithm designed to solve risk-averse optimization problems that incorporate both simulation-based equality constraints and general convex set constraints. The method specifically addresses the computational challenges posed by nonsmooth coherent risk measures, such as Average Value-at-Risk (AVaR). It achieves this by utilizing a risk-averse augmented Lagrangian functional that maintains Fréchet differentiability via epiregularization of the nonsmooth functionals using the previous dual variables in the regularization terms. By integrating this functional with a composite-step trust-region SQP solver, RALES effectively decomposes the problem into manageable components. Simulation-based constraints are handled via inexact, matrix-free linear system solves, while general convex constraints are managed through the augmented Lagrangian. A distinguishing feature of the algorithm is its use of a decoupled penalty parameter update scheme (adapted from the ALESQP framework) during an initial calibration phase of K iterations. This allows the framework to adaptively learn appropriate scalings for heterogeneous sets of constraints. We provide a rigorous convergence analysis for minimizers and dual variables in an ideal setting, and stationary points in a more general, computationally relevant setting. Finally, we illustrate the method's performance and viability through numerical experiments on high-dimensional problems.