  We present a dispersive representation of the $\eta'$ transition form factor
that allows one to account, in a consistent way, for the effects of
$\rho$-$\omega$ mixing in both the isoscalar and the isovector contributions.
Using this formalism, we analyze recent data on $\eta'\to \pi^+\pi^-\gamma$ to
constrain the isovector part of the form factor, individually and in
combination with data for the pion vector form factor. As a first application,
we use our results, in combination with the most recent input for the isoscalar
part of the form factor, to predict the corresponding spectrum of
$\eta'\to\ell^+\ell^-\gamma$, in particular we find the slope parameter
$b_{\eta'}=1.431(23)\, \text{GeV}^{-2}$. With forthcoming data on the latter
process, our results establish the necessary framework to improve the
evaluation of the $\eta'$-pole contribution to the anomalous magnetic moment of
the muon using experimental input from both $\eta'$ decay channels.
