Failure of gauge invariance in the nonperturbative formulation of massless Lorentz-violating QED

We consider a Lorentz-violating modification to the fermionic Lagrangian of QED that is known to produce a finite Chern-Simons term at leading order. We compute the second order correction to the one-loop photon self-energy in the massless case using an exact propagator and a nonperturbative formulation of the theory. This nonperturbative theory assigns a definite value to the coefficient of the induced Chern-Simons term; however, we find that the theory fails to preserve gauge invariance at higher orders. We conclude that the specific nonperturbative value of the Chern-Simons coefficient has no special significance.