  We study a cubic Dirac equation on $\mathbb{R}\times\mathbb{R}^{3}$
\begin{equation*}
  i \partial _t u + \mathcal{D} u + V(x) u =
  \langle \beta u,u \rangle \beta u
  \end{equation*} perturbed by a large potential with almost critical
regularity. We prove global existence and scattering for small initial data in
$H^{1}$ with additional angular regularity. The main tool is an endpoint
Strichartz estimate for the perturbed Dirac flow. In particular, the result
covers the case of spherically symmetric data with small $H^{1}$ norm.
  When the potential $V$ has a suitable structure, we prove global existence
and scattering for \emph{large} initial data having a small chiral component,
related to the Lochak--Majorana condition.
