The hybrid kinematic mechanism (HKM) as a remote handling subsystem of Demonstration Fusion Power Plant (DEMO) breeding blanket (BB) is undergoing extensive theoretical analysis and feasibility verification. In this paper, the forward and inverse kinematic models of HKM are respectively developed by combining the Newtonian iterative method and the analytical method. Cartesian space trajectory planning is performed based on the trajectories of HKM lifting inboard and outboard BBs. The continuous smooth inverse kinematic solutions in HKM joint space are obtained based on the polynomial interpolation method. For the characteristics of HKM piston thread driving, the end-effector position error caused by the degradation of the spherical joint into a universal joint is analyzed and calculated. Combining the overall effects of driving control, rigid-flexible coupling, etc. on position accuracy, an open-loop variable parameter error compensation plan based on the Levenberg-Marquardt (LM) nonlinear damping least squares algorithm is proposed and validated. This study verifies the feasibility of HKM as a BB remote handling subsystem and provides an option for high precision control of HKM.