This paper presents a safety-guaranteed, runtime-efficient imitation learning framework for spacecraft close proximity control. We leverage Control Barrier Functions (CBFs) for safety certificates and Control Lyapunov Functions (CLFs) for stability as unified design principles across data generation, training, and deployment. First, a nonlinear Model Predictive Control (NMPC) expert enforces CBF constraints to provide safe reference trajectories. Second, we train a neural policy with a novel CBF-CLF-informed loss and DAgger-like rollouts with curriculum weighting, promoting data-efficiency and reducing future safety filter interventions. Third, at deployment a lightweight one-step CBF-CLF quadratic program minimally adjusts the learned control input to satisfy hard safety constraints while encouraging stability. We validate the approach for ESA-compliant close proximity operations, including fly-around with a spherical keep-out zone and final approach inside a conical approach corridor, using the Basilisk high-fidelity simulator with nonlinear dynamics and perturbations. Numerical experiments indicate stable convergence to decision points and strict adherence to safety under the filter, with task performance comparable to the NMPC expert while significantly reducing online computation. A runtime analysis demonstrates real-time feasibility on a commercial off-the-shelf processor, supporting onboard deployment for safety-critical on-orbit servicing.