lagrangian formalism - Variation of gamma and Christoffel

I am trying to experiment with Lagrangian densities and came across a term similar to

$$\gamma^i\Gamma^j_{ik}A^k$$ where the $\gamma$ are the gamma matrices, $\Gamma$ are the Christoffel Symbols and $A$ are the field components.

In the variation of the term my question is somewhat twofold: is there any "nice" way to compute $\dfrac{\delta}{\delta g^{lm}}\gamma^a$, knowing the relation $\{\gamma^i,\gamma^j\}=2g_{ij}\mathbf{I}$?

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Secondly, from what I have seen $\delta\Gamma^i_{jk}=\Gamma_{jkl}\delta g^{il}+g^{il}\delta\Gamma_{jkl}$, commonly rewritten in a different form. The question Variation of the Christoffel symbols with respect to $g^{\mu\nu}$ does not seem to have a satisfactory answer for my purpose; it seems like the variation terms will stay in the field equation so it does not make sense to leave variations in the expression. Is it valid to naively assume that $\dfrac{\delta}{\delta g^{lm}}\Gamma^j_{ik}=\Gamma_{jkl}$?