Fermionic Casimir effect in an axial Lorentz-violating background

We investigate the fermionic Casimir effect for a Dirac field confined between two parallel plates with MIT bag boundary conditions in the presence of CPT-odd Lorentz-symmetry violation described by a constant axial background vector $b_{\mu}$. The exact mode quantization is derived from the modified Dirac equation in the planar geometry, and the vacuum energy is formulated through a phase-shift representation. For spacelike backgrounds we show that the components parallel to the plates can be absorbed into a shift of the transverse momenta and therefore do not affect the renormalized Casimir energy, while the component normal to the plates modifies the longitudinal spectrum and produces a genuine Lorentz-violating correction. Both the timelike component $b_{0}$ and the normal spacelike component $b_{z}$ can thus be treated within a unified framework characterized by a single effective spectral parameter. A closed logarithmic integral representation for the Casimir energy is obtained and its behavior is analyzed in the Lorentz-symmetric, weak-background, and strong-background regimes.