This work presents a conservative treatment of solid boundaries that enforces strict mass conservation and mass-energy flux consistency at solid boundaries while maintaining the efficiency of the hybrid Lattice-Boltzmann/Finite-Volume (LB/FV) method for simulating compressible fluid flows. In this method, the mass and momentum equations are solved using the hybrid recursive regularized LB (HRR) algorithm, while the total energy equation is solved using a FV scheme. The strategy proposed by Zhao et al. ( 2020) is used to strongly couple LB mass and FV energy fluxes by evaluating the latter directly from the distribution functions of the LBM kinetic part. Furthermore, a modified regularized boundary condition is introduced to reconstruct the pre-collision populations at the boundaries. At boundary nodes, the density is updated from a lattice-consistent continuity balance, in which the mass variation of the boundary control volume is computed from the post-collision population exchange along the available fluid links, while zero mass exchange is imposed across blocked solid links. A series of validation cases involving conservation tests, thermal wall conditions, wall-force evaluation, and compressible-flow benchmarks demonstrates the effectiveness of the proposed boundary treatment in restoring the targeted discrete conservation properties without degrading the prediction of relevant wall and aerodynamic quantities.