  We show that the average size of the $2$-Selmer group of the family of
Jacobians of non-hyperelliptic genus-$3$ curves with a marked rational
hyperflex point, when ordered by a natural height, is bounded above by $3$. We
achieve this by interpreting $2$-Selmer elements as integral orbits of a
representation associated with a stable $\mathbb{Z}/2\mathbb{Z}$-grading on the
Lie algebra of type $E_6$ and using Bhargava's orbit-counting techniques. We
use this result to show that the marked point is the only rational point for a
positive proportion of curves in this family. The main novelties are the
construction of integral representatives using certain properties of the
compactified Jacobian of the simple curve singularity of type $E_6$, and a
representation-theoretic interpretation of a Mumford theta group naturally
associated to our family of curves.
