We present a design procedure for low-profile single-feed cavity-excited omega-bianisotropic metasurface antennas (CX-OBMAs), capable of generating an arbitrary desirable field distribution on their aperture. The procedure relies on a greedy algorithm that optimizes the source position inside the cavity and the magnitude and phase of the metasurface modal reflection coefficients, such that the power profile generated by the cavity fields matches the one corresponding to the specified aperture distribution. Once this local power conservation is established, we invoke previously derived analytical formulas to evaluate the (passive lossless) metasurface constituents that facilitate the required field transformation. We demonstrate this scheme by designing CX-OBMAs that generate radiation patterns with prescribed directivity and side-lobe level, for which the aperture fields can be readily stipulated using established antenna array theory. The good agreement between full-wave simulations and semianalytical predictions points out the immense potential of these novel radiators, capable of controlling near- and far-field distributions almost at will. In particular, they can emulate antenna array performance, without requiring design and implementation of complicated, expensive, and lossy feed networks.