In this paper we study the transitions of entanglement complexity in an exemplary family of states - the Rokhsar-Kivelson-sign wavefunctions - whose degree of entanglement is controlled by a single parameter. This family of states is known to feature a transition between a phase exhibiting volume-law scaling of entanglement entropy and a phase with sub-extensive scaling of entanglement, reminiscent of the many-body-localization transition of disordered quantum Hamiltonians [Physical Review B 92, 214204 (2015)]. We study the singularities of the RK-sign wavefunctions and their entanglement complexity across the transition using several tools from quantum information theory: fidelity metric; entanglement spectrum statistics; entanglement entropy fluctuations; stabilizer Rényi Entropy; and the performance of a disentangling algorithm. Across the whole volume-law phase the states feature universal entanglement spectrum statistics. Yet a "super-universal" regime appears for small values of the control parameter in which all metrics become independent of the parameter itself; the entanglement entropy as well as the stabilizer Rényi entropy appear to approach their theoretical maximum; the entanglement fluctuations scale to zero as in output states of random universal circuits, and the disentangling algorithm has essentially null efficiency. All these indicators consistently reveal a complex pattern of entanglement. In the sub-volume-law phase, on the other hand, the entanglement spectrum statistics is no longer universal, entanglement fluctuations are larger and exhibiting a non-universal scaling; and the efficiency of the disentangling algorithm becomes finite. Our results, based on model wavefunctions, suggest that a similar combination of entanglement scaling properties and of entanglement complexity features may be found in high-energy Hamiltonian eigenstates.