Background to my question: the flat case
I am interested in the following (part of a) Lagrangian involving spinors in a general curved spacetime: $$L=\frac{1}{2}\overline{\psi}\left(i\gamma^{\alpha}\nabla_{\alpha}-\mu_{0}\right)\psi$$ where the covariant derivative acting on fermions is defined using the vielbeins (as described in S. Weinberg, Gravitation and Cosmology, John Wiley&Sons, New York, 1972):
$$\nabla_{\alpha}=V_{\alpha}^{\,\,\,\,\mu}\partial_{\mu}+\frac{1}{2}\sigma^{\beta\gamma}V_{\beta}^{\,\,\,\,\nu}V_{\alpha}^{\,\,\,\,\mu}\left(\partial_{\mu}V_{\gamma\nu}-\Gamma_{\nu\mu}^{\lambda}V_{\gamma\lambda}\right)$$ where $$\sigma^{\alpha\beta}=\frac{1}{4}\left[\gamma^{\alpha},\gamma^{\beta}\right]$$
To find the energy-momentum tensor (EMT) associated with this Lagrangian, I used the following relation (derived from the standard definition of the EMT, along with the relation $g_{\mu\nu}\left(x\right)=V_{\,\,\,\,\mu}^{\alpha}\left(x\right)V_{\,\,\,\,\nu}^{\beta}\left(x\right)\eta_{\alpha\beta}$):
$$T_{\mu\nu}=-g_{\mu\nu}L-g_{\mu\rho}V_{\,\,\,\,\nu}^{\beta}\left(x\right)\frac{\delta L}{\delta V_{\,\,\,\,\rho}^{\beta}\left(x\right)}$$
I began by finding the EMT in flat spacetime, by varying around Minkowski (that is, taking $V_{\,\,\,\,\alpha}^{\beta}=\delta_{\,\,\,\,\alpha}^{\beta}+\epsilon_{\,\,\,\,\alpha}^{\beta}$ and expanding to order $\epsilon$). Using various manipulations, identities on $\gamma$ and $\sigma$, and the equations of motion, I obtained:
$$T_{\mu\nu}^\flat= \frac{i}{8}\overline{\psi}\left(\gamma_{\mu}\overrightarrow{\partial}_{\nu}+\gamma_{\nu}\overrightarrow{\partial}_{\mu}\right)\psi-\frac{i}{8}\overline{\psi}\left(\gamma_{\mu}\overleftarrow{\partial}_{\nu}+\gamma_{\nu}\overleftarrow{\partial}_{\mu}\right)\psi -\eta_{\mu\nu}\left(\frac{1}{2}\overline{\psi}i\gamma^{\alpha}\partial_{\alpha}\psi-\frac{\mu_{0}}{2}\overline{\psi}\psi\right) $$
where $\flat$ denotes flat spacetime.
My question: generalizing this result to curved spacetime
My original aim was to find the EMT in curved spacetime. The long way would be to return to the definition of $T_{\mu\nu}$ as I wrote above, and perform the full derivation in curved spacetime. But having already computed $T_{\mu\nu}^\flat$, I turned to the simple solution: Take $T_{\mu\nu}^\flat$ (as given above), write it in a fully covariant manner (which is easy: just replace $\partial_\mu$ with $\nabla_\mu$ and $\eta_{\mu\nu}$ with $g_{\mu\nu}$). Then, since it is a covariant equation written in one coordinate system (local Cartesian coordinates, in which the flat and the covariant expressions coincide), it may be transformed to hold in any coordinate system, yielding the general expression in curved spacetime.
This results in:
$$T_{\mu\nu}= \frac{i}{8}\overline{\psi}\left(\gamma_{\mu}\overrightarrow{\nabla}_{\nu}+\gamma_{\nu}\overrightarrow{\nabla}_{\mu}\right)\psi-\frac{i}{8}\overline{\psi}\left(\gamma_{\mu}\overleftarrow{\nabla}_{\nu}+\gamma_{\nu}\overleftarrow{\nabla}_{\mu}\right)\psi -g_{\mu\nu}\left(\frac{1}{2}\overline{\psi}i\gamma^{\alpha}\nabla_{\alpha}\psi-\frac{\mu_{0}}{2}\overline{\psi}\psi\right) $$
This is indeed the correct expression for the covariant EMT of the given Lagrangian.
My question is, is my reasoning valid or does it require further justification? E.g., can't there be extra terms in the covariant expression that vanish in flat spacetime, hence not allowing the full construction of $T_{\mu\nu}$ from $T_{\mu\nu}^\flat$? What guarantees there are no such terms in this case?
