In Cooperative Intelligent Transport Systems (C-ITS), vehicles can disseminate announcements to the traffic Application Server (AS) and nearby vehicles to enhance traffic safety and efficiency. To build reliable dissemination environments, various threshold announcement approaches are proposed, wherein receivers accept a traffic event only when t (threshold) vehicles have endorsed it. Among these approaches, group signaturebased approaches have emerged as promising choices for C-ITS due to their optimal tradeoff among certificate management, privacy protection, and traceability. However, existing group signature-based approaches have three limitations: 1) Announcement verification is time-consuming, making them unable to meet the fast authentication requirement of C-ITS. 2) Receivers face significant storage burdens in maintaining numerous traffic events and endorsements. 3) Unsuitable for vehicle-to-AS announcement dissemination as their one-by-one dissemination mode would impose unaffordable costs to the central AS, which limits their applicability in certain scenarios. To overcome these issues, we design an efficient group signature scheme tailored to reliable anonymous announcement dissemination. Building upon this scheme, we further present a novel threshold anonymous announcement protocol, which optimizes the announcement's dissemination mode and enables efficient batch verification. Specifically, instead of the traditional one-by-one dissemination, a compressed anonymous announcement with t endorsements is disseminated to receivers. This dissemination mode facilitates immediate announcement processing, eliminates the need for extensive traffic information maintenance, and is suitable for both Vehicle-to-Vehicle (V2V) and vehicle-to-AS announcement dissemination scenarios. Meanwhile, regardless of the threshold t, the announcement can be verified with only a small constant number of pairing and scalar multiplication operations. The theoretical analysis demonstrates that our protocol achieves the expected properties, and extensive simulation results show that our protocol outperforms the state-of-the-art protocols, indicating the performance and practicability of our protocol.