We consider the class of nonsmooth regularized optimization problems with equality constraints. Such problems arise in signal processing, machine learning, and low-rank matrix completion, where the nonsmooth term promotes structural properties such as sparsity or low total variation. We propose an extension of the Sequential Quadratic Programming (SQP) method introduced by Curtis, Qu and Robinson (2026) that handles a broad class of nonsmooth regularizers beyond the \(\ell_1\) norm. At each iteration, the tangent subproblem is solved using the Envelopt solver combined with a null-space reduction, which exploits the Moreau envelope to handle the nonsmooth term while enforcing the linearized constraints exactly. We further show that solving the tangent subproblem inexactly preserves the convergence and complexity results, provided that simple and sufficient decrease conditions are satisfied. Numerical experiments on problems from the CUTEst library confirm that the proposed method successfully handles various convex regularizers, and that inexact tangent step resolution significantly reduces the total number of inner solver iterations while maintaining solution quality.