First introduced decades ago, the Short-Backfire Antenna (SBFA) has proven to be a well-defined, reliable, and rugged radiator often selected for use in harsh environments requiring gain greater than 15 dBi. In this report, we present the optimization of a circular Advanced Short Backfire Antenna (A-SBF A) which has been designed for dual-band operation with remarkably high aperture efficiency over a range of different aperture sizes. In contrast to the hard-EM SBFA, anisotropic, dispersive metasurfaces line the cavity walls of the circular A-SBF A and prove integral to achieving 100% aperture efficiency at the two frequency bands for diameters spanning a range from 1.0λ_0 up to 2.13λ_0. We present an inverse-design methodology used to optimize such high-performing A-SBFA antennas, and the pursuant results. In addition, we draw a comparison between the circular A-SBFA and traditional high-efficiency short horn antennas, which is illustrative of the great impact of the circular A-SBFA.