Rename is an option for Contract. If set to True, dummy indices in Eps are renamed, using $MU[i].
LC[\[Mu], \[Nu], \[Rho], \[Sigma]] LC[\[Alpha], \[Nu], \[Rho], \[Sigma]]
Contract[%, EpsContract -> False, Rename -> True]
\bar{\epsilon }^{\alpha \nu \rho \sigma } \bar{\epsilon }^{\mu \nu \rho \sigma }
\bar{\epsilon }^{\alpha \;\text{\$MU}(4)\text{\$MU}(5)\text{\$MU}(6)} \bar{\epsilon }^{\mu \;\text{\$MU}(4)\text{\$MU}(5)\text{\$MU}(6)}