Understanding critical phenomena is of central importance to condensed-matter and high-energy physics. Such an understanding is reflected in our ability to sort observables based on their degeneracy, symmetries, and power-law decays. Here, we study such critical properties of the Ising-Higgs gauge theory in along the self-dual line which have recently been a subject of debate. Using machine learning techniques, we determine the low-energy operator content of the associated field theory. Our approach enables us to largely refute the existence of an emergent current operator and with it the standing conjecture that this transition is of the universality class. We contrast these results with the ones obtained for the Ashkin-Teller transverse field Ising model where we find the expected current operator. Our numerical technique extends the recently proposed real-space mutual information allowing us to extract subleading nonlinear operators. This allows a controlled and computationally scalable approach to target the conformal field theory spectrum. and discern universality classes beyond from Monte Carlo data.