We show how the length scale hierarchy, resulting from different interaction strengths in an optically-trapped spin-1 Na-23 Bose-Einstein condensate, can lead to intriguing core deformations in singular topological defects. In particular, a point defect can be unstable with respect to the formation of a stable half-quantum vortex ring (an 'Alice ring'), providing a realistic scheme to use dissipation as a sophisticated state engineering tool. We compute the threshold for stability of the point monopole, which is beyond the current experimental regime.