  In this paper, we apply the form factor bootstrap approach to branch point
twist fields in the $q$-state Potts model for $q\leq 3$. For $q=3$ this is an
integrable interacting quantum field theory with an internal discrete
$\mathbb{Z}_3$ symmetry and therefore provides an ideal starting point for the
investigation of the symmetry resolved entanglement entropies. However, more
generally, for $q\leq 3$ the standard R\'enyi and entanglement entropies are
also accessible through the bootstrap programme. In our work we present form
factor solutions both for the standard branch point twist field with $q\leq 3$
and for the composite (or symmetry resolved) branch point twist field with
$q=3$. In both cases, the form factor equations are solved for two particles
and the solutions are carefully checked via the $\Delta$-sum rule. Using our
analytic predictions, we compute the leading finite-size corrections to the
entanglement entropy and entanglement equipartition for a single interval in
the ground state.
