In this paper, we present a novel cooperative scheme of joint optimal resource allocation, such that the overall performance of the coexisting radar-communications (CRC) system can be improved. In our proposed scheme, target detection and multiuser communication are performed by radar and communication subsystems at the same time, as well as a control center, which is responsible for joint resource management. We aim to minimize the ISLR for target detection and maximize the sum-rate for communications simultaneously by jointly optimizing the band assignment and transmit power allocation. Since the resulting optimization problem involving two performance metrics and a binary constraint is a multiobjective nonconvex problem, a two-tier iterative decomposition (TT-ID) approach is devised to obtain the globally optimal solution. However, compared with the conventional radar signals, the autocorrelation function of the devised radar signal may still have relatively high sidelobes. In particular, when the data transmission becomes the primary purpose of the CRC system, the sidelobe performance gets worse. As a consequence, some weak targets are most likely overshadowed by the adjacent strong targets through the matched filtering at the radar receiver. To address this, a spectral estimation algorithm based on the Bayes Cauchy–Gaussian (Bayes–CG) model is employed to further reduce the range sidelobes of the matched filter output at the radar receiver according to the prior distribution of the desired autocorrelation. Finally, several numerical results are provided to show the merits of the proposed method.