  We show an analogue of the Lorentzian splitting theorem for weighted
Lorentz-Finsler manifolds: If a weighted Berwald spacetime of nonnegative
weighted Ricci curvature satisfies certain completeness and metrizability
conditions and includes a timelike straight line, then it necessarily admits a
one-dimensional family of isometric translations generated by the gradient
vector field of a Busemann function. Moreover, our formulation in terms of the
$\epsilon$-range introduced in our previous work enables us to unify the
previously known splitting theorems for weighted Lorentzian manifolds by Case
and Woolgar-Wylie into a single framework.
