Very interesting discussion, and I would like to put there my version of this problem.
First, all identities only have any sense if basic conventional points are postulated. Namely, below I will use the following definition of four-dimensional Levi-Civita symbol:
$$
\varepsilon^{\alpha\beta\gamma\delta}=-\varepsilon_{\alpha\beta\gamma\delta}=
\begin{cases}
+1,& \text{if } \alpha\beta\gamma\delta=\text{even permutation of 0123}\\
-1,& \text{if } \alpha\beta\gamma\delta=\text{odd permutation of 0123}.
\end{cases}
$$
Next, the definition of fifth gamma-matrix is the following:
$$
\gamma^5=\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3=-i\gamma_0\gamma_1\gamma_2\gamma_3.
$$
With such definitions
$$
\mathrm{Tr}\{{\gamma^5\gamma^{\alpha}\gamma^{\beta}\gamma^{\gamma}\gamma^{\delta}}\}=\mathrm{Tr}\{{\gamma^{\alpha}\gamma^{\beta}\gamma^{\gamma}\gamma^{\delta}\gamma^5}\}=-4i\varepsilon^{\alpha\beta\gamma\delta},\tag{1}\label{1}
$$
what is checked for the component $\alpha=0,\ \beta=1,\ \gamma=2,\ \delta=3$ and is carried over to the other components which are permutations of 0123.
Further, there is the identity
$$
\gamma^{\alpha}\gamma^{\beta}\gamma^{\gamma}=\eta^{\alpha\beta}\gamma^{\gamma}-\eta^{\alpha\gamma}\gamma^{\beta}+\eta^{\beta\gamma}\gamma^{\alpha}-i\gamma^5\varepsilon^{\alpha\beta\gamma\delta}\gamma_{\delta},\tag{2}\label{2}
$$
where $\eta^{\mu\nu}$ is the Minkowski metric with the signature $\{+1, -1, -1, -1\}$ and, hence, $\gamma_0=\gamma^0$, while $\gamma_\mu=-\gamma^\mu$ with $\mu=\{1, 2, 3\}$. Such defined the identity is in agreement with the version of @J.G. because $\gamma^5$ anticommutes with any of $\gamma_\mu$ and $\varepsilon^{\delta\alpha\beta\gamma}=-\varepsilon^{\alpha\beta\gamma\delta}$.
Finally, we are ready to calculate the trace $\mathrm{Tr}\{\gamma^{\alpha}\gamma^{\beta}\gamma^{\gamma}\gamma^{\rho}\gamma^{\sigma}\gamma^{\tau}\gamma^5\}$.
First, replacing $\gamma^{\alpha}\gamma^{\beta}\gamma^{\gamma}$ with the use of the identity \ref{2} one obtains
$$
\mathrm{Tr}\{\gamma^\alpha\gamma^\beta\gamma^\gamma\gamma^\rho\gamma^\sigma\gamma^\tau\gamma^5\}=
\eta^{\alpha\beta}\mathrm{Tr}\{\gamma^\gamma\gamma^\rho\gamma^\sigma\gamma^\tau\gamma^5\}-
\eta^{\alpha\gamma}\mathrm{Tr}\{\gamma^\beta\gamma^\rho\gamma^\sigma\gamma^\tau\gamma^5\}+
\eta^{\beta\gamma}\mathrm{Tr}\{\gamma^\alpha\gamma^\rho\gamma^\sigma\gamma^\tau\gamma^5\}-
i\varepsilon^{\alpha\beta\gamma\delta}\mathrm{Tr}\{\gamma^5\gamma_\delta\gamma^\rho\gamma^\sigma\gamma^\tau\gamma^5\}.\tag{3}\label{3}
$$
First three terms are rewritten with the use of the identity \ref{1}.
As for the last term, let's consider the multiplier $\mathrm{Tr}\{\gamma^5\gamma_\delta\gamma^\rho\gamma^\sigma\gamma^\tau\gamma^5\}$ separately:
$$
\mathrm{Tr}\{\gamma^5\gamma_\delta\gamma^\rho\gamma^\sigma\gamma^\tau\gamma^5\}=\mathrm{Tr}\{\gamma_\delta\gamma^\rho\gamma^\sigma\gamma^\tau\}
$$
because of anticommutation of $\gamma^5$ with both $\gamma_\mu$ and $\gamma^\mu$, further use the identity \ref{2} one more time:
$$
\mathrm{Tr}\{\gamma_\delta\gamma^\rho\gamma^\sigma\gamma^\tau\}=
\eta^{\rho\sigma}\mathrm{Tr}\{\gamma_\delta\gamma^\tau\}-
\eta^{\rho\tau}\mathrm{Tr}\{\gamma_\delta\gamma^\sigma\}+
\eta^{\sigma\tau}\mathrm{Tr}\{\gamma_\delta\gamma^\rho\}-
i\varepsilon^{\rho\sigma\tau\eta}\mathrm{Tr}\{\gamma_\delta\gamma^5\gamma_\eta\}.\tag{4}\label{4}
$$
The last term vanishes (trace of product of fifth gamma-matrix with only two gamma-matrices) and every of the other terms contains the multiplier of the form
$$
\mathrm{Tr}\{\gamma_\lambda\gamma^\nu\}=\eta_{\lambda\mu}\mathrm{Tr}\{\gamma^\mu\gamma^\nu\}=4\eta_{\lambda\mu}\eta^{\mu\nu}.
$$
Using this one can rewrite \ref{4} in the form
$$
\mathrm{Tr}\{\gamma_\delta\gamma^\rho\gamma^\sigma\gamma^\tau\}=
4\eta^{\rho\sigma}\eta_{\delta\lambda}\eta^{\lambda\tau}-
4\eta^{\rho\tau}\eta_{\delta\lambda}\eta^{\lambda\sigma}+
4\eta^{\sigma\tau}\eta_{\delta\lambda}\eta^{\lambda\rho}
$$
and inserting it to \ref{3} one obtains
$$
\mathrm{Tr}\{\gamma^\alpha\gamma^\beta\gamma^\gamma\gamma^\rho\gamma^\sigma\gamma^\tau\gamma^5\}=-4i(
\eta^{\alpha\beta}\varepsilon^{\gamma\rho\sigma\tau}
-\eta^{\alpha\gamma}\varepsilon^{\beta\rho\sigma\tau}
+\eta^{\beta\gamma}\varepsilon^{\alpha\rho\sigma\tau}
+\eta^{\rho\sigma}\varepsilon^{\alpha\beta\gamma\tau}
-\eta^{\rho\tau}\varepsilon^{\alpha\beta\gamma\sigma}
+\eta^{\sigma\tau}\varepsilon^{\alpha\beta\gamma\rho}).
$$
Note, that we have used the following transformations:
$$
\varepsilon^{\alpha\beta\gamma\delta}\eta_{\delta\lambda}\eta^{\lambda\tau}=
\varepsilon^{\alpha\beta\gamma}_{\phantom{\alpha\beta\gamma}\lambda}\eta^{\lambda\tau}=
\varepsilon^{\alpha\beta\gamma\tau}
$$
and so on.
