Rare events, despite their infrequency, often carry critical information and require immediate attentions in mission-critical applications such as autonomous driving, healthcare, and industrial automation. The data-intensive nature of these tasks and their need for prompt responses, combined with designing edge AI (or edge inference), pose significant challenges in systems and techniques. Existing edge inference approaches often suffer from communication bottlenecks due to high-dimensional data transmission and fail to provide timely responses to rare events, limiting their effectiveness for mission-critical applications in the sixth-generation (6G) mobile networks. To overcome these challenges, we propose a channel-adaptive, event-triggered edge-inference framework that prioritizes efficient rare-event processing. Central to this framework is a dual-threshold, multi-exit architecture, which enables early local inference for rare events detected locally while offloading more complex rare events to edge servers for detailed classification. To further enhance the system's performance, we developed a channel-adaptive offloading policy paired with an online algorithm to dynamically determine the optimal confidence thresholds for controlling offloading decisions. The associated optimization problem is solved by reformulating the original non-convex function into an equivalent strongly convex one. Using deep neural network classifiers and real medical datasets, our experiments demonstrate that the proposed framework not only achieves superior rare-event classification accuracy, but also effectively reduces communication overhead, as opposed to existing edge-inference approaches.