  Motivated by a simultaneous explanation of the apparent discrepancies in the
light charged lepton anomalous magnetic dipole moments, and the anomalous
internal pair creation in $^8$Be nuclear transitions, we explore a simple New
Physics model, based on an extension of the Standard Model gauge group by a
$U(1)_{B-L}$. The model further includes heavy vector-like fermion fields, as
well as an extra scalar responsible for the low-scale breaking of $U(1)_{B-L}$,
which gives rise to a light $Z^\prime$ boson. The new fields and currents allow
to explain the anomalous internal pair creation in $^8$Be while being
consistent with various experimental constraints. Interestingly, we find that
the contributions of the $Z^\prime$ and the new $U(1)_{B-L}$-breaking scalar
can also successfully account for both $(g-2)_{e,\mu}$ anomalies; the strong
phenomenological constraints on the model's parameter space ultimately render
the combined explanation of $(g-2)_e$ and the anomalous internal pair creation
in $^8$Be particularly predictive. The underlying idea of this minimal
"prototype model" can be readily incorporated into other protophobic $U(1)$
extensions of the Standard Model.
