Accurate camera-to-robot calibration is essential for any vision-based robotic control system and especially critical in minimally invasive surgical robots, where instruments conduct precise micro-manipulations. However, MIS robots have long kinematic chains and partial visibility of their degrees of freedom in the camera, which introduces challenges for conventional camera-to-robot calibration methods that assume stiff robots with good visibility. Previous works have investigated both keypoint-based and rendering-based approaches to address this challenge in real-world conditions; however, they often struggle with consistent feature detection or have long inference times, neither of which are ideal for online robot control. In this work, we propose a novel framework that unifies the detection of geometric primitives (keypoints and shaft edges) through a shared encoding, enabling efficient pose estimation via projection geometry. This architecture detects both keypoints and edges in a single inference and is trained on large-scale synthetic data with projective labeling. This method is evaluated across both feature detection and pose estimation, with qualitative and quantitative results demonstrating fast performance and state-of-the-art accuracy in challenging surgical environments.