EpsEvaluate[expr] applies total antisymmetry and linearity (w.r.t. momenta) to all Levi-Civita tensors (Eps) in expr.
Overview, Contract, Eps, LC, Trick.
Contract[LC[\[Mu], \[Nu], \[Rho], \[Sigma]] FV[p + q, \[Sigma]]] // MomentumCombine
ex = EpsEvaluate[%]\bar{\epsilon }^{\mu \nu \rho \overline{p}+\overline{q}}
\bar{\epsilon }^{\mu \nu \rho \overline{p}}+\bar{\epsilon }^{\mu \nu \rho \overline{q}}
ex // StandardForm
(*Eps[LorentzIndex[\[Mu]], LorentzIndex[\[Nu]], LorentzIndex[\[Rho]], Momentum[p]] + Eps[LorentzIndex[\[Mu]], LorentzIndex[\[Nu]], LorentzIndex[\[Rho]], Momentum[q]]*)