In this study, inextensible flows of curves in pseudo-Galilean 4-space were comprehensively characterized, and the necessary and sufficient conditions governing these curve flows were established as a coupled system of partial differential equations. By defining the directional derivatives with respect to the parameter $t$ and the arc-length parameter $s$ in accordance with the Serret-Frenet frame, we derived the extended Serret-Frenet relations specifically adapted to the metric in $G_1^4$. This formulation ensured the preservation of the curve's arc length during deformation, effectively modeling the motion of inelastic filaments. Beyond the kinematic description, we utilized the Sasaki metric framework on the tangent bundle to formulate the bending elastic energy functionals and pseudo-angle variations for the moving frame fields. These energy profiles were evaluated along both spatial $s-$ lines and dynamic $t-$ lines, explicitly demonstrating how geometric constraints and metric indices differentiated the system's energy distribution and energy transfer mechanisms. Furthermore, the structural integrity of the flow was evaluated through Lyapunov stability criteria, proving that the inextensibility constraint stabilized the higher-dimensional variations and rendered the system orbitally stable. The framework was validated via helicoidal trajectory simulations, aligning derived equations with physical thresholds. This work provided a rigorous mathematical basis for inelastic rod and fiber dynamics in 4D pseudo-Galilean space.