We study the existence of self-dual solitons in a gauged version of the baby Skyrme model in which the Maxwell-Chern-Simons term governs the gauge field dynamic. For such a purpose, a detailed implementation of the Bogomol'nyi-Prasad-Sommerfield formalism provides a lower bound for the energy and the respective self-dual equations. The energy lower bound is quantized because it is proportional to the topological charge of the Skyrme field. Furthermore, neither the magnetic flux nor the electric charge is quantized.We find two types of self-dual Skyrme field profiles: the first is described by a solution which follows a Gaussian decay law, and the second is portrayed by a solution having a power-law decay. For both types of skyrmion profiles, the asymptotic behavior of the respective gauge field is similar to the one for the Abrikosov-Nielsen-Olesen vortices. The localized inversion of the magnetic flux is another feature not observed in other gauged Skyrme models already studied in the literature.