Light beams possess two intrinsic quantized degrees of freedom, related to spin angular momentum (SAM) and orbital angular momentum (OAM), whose manipulation enables extensive control over the topological properties of electromagnetic fields. In this context, structured fields constructed from a non-separable combination of SAM and OAM have recently gained sustained interest. Such states are eigenstates of the so-called generalized angular momentum (GAM), a mixed angular momentum operator encompassing both SAM and OAM components, which can result in astonishing fractional eigenvalues. The demonstration of GAM conservation under harmonic generation has suggested a potential relevance of this new form of angular momentum as a meaningful quantum number. In the present work, we expand the scope of evaluation by investigating its conservation law with second-harmonic generation in an underdense isotropic inhomogeneous plasma that relies on dipole-forbidden interaction implying spin-orbit coupling. Our study reveals that the symmetry and topological properties of the field are disrupted during the nonlinear process, the GAM charge being only conserved on average. This symmetry breaking can be exploited to provide an easily detectable signature of the driving field topology or to create a robust topological attractor.