The one-loop neutrino mass models based on the topological features of the dimension-5 Weinberg operator have been systematically categorized into three categories. Notably, within these topological categories, the extension of canonical seesaw scenarios at the one-loop level is of interest given the current LHC run. Besides the one-loop contribution, these extensions yield a prevalent tree-level contribution to neutrino masses. Achieving a dominant one-loop contribution necessitates the amalgamation of flavor symmetries and an expanded field content. Alternatively, we propose the realization of a specific topological Lorentz structure (T4-2-i) associated with the one-loop extension of Type-II seesaw employing modular A4 symmetry. In this realization, no additional fields are required beyond those allowed by the topology itself. The modular weights play a crucial role in suppressing tree-level terms and stabilizing the particles participating in the loop (such as \(N_i\) , \(\rho \) , and \(\phi \) ), rendering them potential dark matter candidates.