This paper analyzes the integration of Non-Orthogonal Multiple Access (NOMA) in a Fog Radio Access Network (FRAN) architecture with limited fronthaul capacity. More precisely, it proposes methods for optimizing the resource allocation for the downlink of a NOMA-based FRAN with multiple resource blocks (RB). The resource allocation problem is formulated as a mixed-integer optimization problem, which determines the user-to-RB assignment, the power allocated to each RB, and the power split levels of the NOMA users served by each RB. The optimization problem maximizes a networkwide rate-based utility function subject to fronthaul-capacity constraints. The paper proposes a feasible decoupled solution for such a non-convex optimization problem using a three-step hybrid centralized/distributed approach, which in part relies on the edge-devices computation capabilities. The paper proposes and compares two distinct methods for solving the assignment problem, namely a Hungarian-based method, and a Multiple Choice Knapsack-based method. The power allocation to RBs and the NOMA power split optimization are solved using the alternating direction method of multipliers (ADMM). Simulations results illustrate the advantages of the proposed methods compared to different baseline schemes, including the conventional Orthogonal Multiple Access (OMA), for different utility functions and different network environments.