This paper proposes an integrated sensing and communication (ISAC) system covert waveform design method for complex clutter environments, with the core objective of maximizing the signal-to-clutter-plus-noise ratio (SCNR). The design achieves efficient clutter suppression while meeting the covertness requirement through joint optimization of the transmit waveform and receive filter, enabling cooperative radar detection and wireless communication. This study presents key innovations that explicitly address target Doppler shift uncertainty, significantly enhancing system robustness against Doppler effects. To ensure communication reliability, the method incorporates phase difference constraints between communication signal elements in the waveform design, along with energy constraint, covert constraint, and peak-to-average power ratio (PAPR) constraint. The original non-convex optimization problem is transformed into a tractable convex optimization form through convex optimization technique. Simulation results demonstrate that the optimized waveform not only satisfies the covertness requirement in complex clutter environment, but also achieves superior target detection performance. It also ensures reliable communication and confirms the effectiveness of propose method.