  Lorentz violation through a radiatively induced Chern-Simons-like term in a
fermionic theory embedded in linearized quantum gravity with a Lorentz- and
CPT-violating axial-vector term in the fermionic sector proportional to a
constant field $b_\mu$ has been recently studied. In a similar fashion as for
the extended-QED model of Carroll-Field-Jackiw, we explicitly show that neither
gauge invariance nor the more stringent momentum routing invariance condition
on underlying Feynman diagrams fix the arbitrariness inherent to such induced
term at one loop order. We present the calculation in a nonperturbative
expansion in $b_\mu$ and within a framework which besides operating in the
physical dimension (and thus avoiding $\gamma_5$ matrix Clifford algebra
ambiguities), judiciously parametrizes regularization dependent arbitrary
parameters usually fixed by symmetries.
