This paper presents a projection-error based backstepping control (BSC) framework for a three-phase, two-level grid-connected rectifier to ensure optimum DC-link voltage regulation operating under discrete-valued input constraints. While continuous-time backstepping ensures accurate regulation of the DC-link voltage and inner-loop current dynamics, realworld implementations necessitate control inputs drawn from a finite set due to switching and quantization limitations. To address this, we project the nominal control law onto a finite control set and rigorously analyze the impact of this projection on closed-loop stability. A Lyapunov-based proof is developed to establish stability under bounded projection errors, with explicit sufficient conditions relating the quantization error and controller gains. Theoretical results are supported by simulation studies demonstrating robust DC voltage regulation and smooth grid current tracking under DC-link voltage reference variation and unity power factor. The proposed methodology is validated through detailed simulations and hard-ware-in-the-loop (HIL) experimental results.