To address the challenges of low positioning accuracy and poor robustness caused by the strong nonlinear coupling of piezoelectric hysteresis, mechanical friction, and transmission gaps in a macro-micro composite-driven cell injection mechanism, an Adaptive Fast Non-singular Terminal Sliding Mode (FNTSM) control strategy is proposed. This strategy aims to eliminate the reliance on an accurate dynamic model and achieve precise motion control across scales. First, a dynamic model of the piezoelectric-driven cell injection mechanism is established, considering uncertainties such as hysteresis, friction, and other perturbations. Next, to overcome the traditional sliding mode control's dependency on prior knowledge of disturbance bounds and its chattering issue, an FNTSM controller (APIDSM-TDE) is designed by integrating time delay estimation (TDE) technology and adaptive gain adjustment. The controller uses TDE to estimate and compensate for lumped system disturbances online and guarantees the system state convergence in finite time through FNTSM. At the same time, a PID-type sliding surface is introduced, and an adaptive law is designed to dynamically adjust the sliding surface parameters, enhancing the system's ability to suppress strong time-varying disturbances, such as varying friction coefficients and sudden load changes, while improving the response speed. Experimental results show that, compared with traditional PID controllers, the APIDSM-TDE controller reduces the maximum tracking error percentage by 62.8% and the root mean square error percentage by 15.8%. The controller effectively improves the trajectory tracking accuracy of the cell injection mechanism in cross-scale motion, providing a reliable technical solution for micro-manipulation.