  We extend our studies of a new class of $2 \to 3$ exclusive processes using
the collinear factorisation framework by considering the exclusive
photoproduction of a $\gamma\,\rho$ pair, in the kinematics where the pair has
a large invariant mass, {and the outgoing $ \rho $-meson has a sufficiently
large transverse momentum to not resonate with the nucleon}. We cover the whole
kinematical range from medium energies in fixed target experiments to very
large energies of colliders, by considering the experimental conditions of
JLab~12-GeV, COMPASS, future EIC and LHC (in ultra-peripheral collisions)
cases. One of the main interest in studying the present process is that it
provides access to both chiral-even and chiral-odd GPDs, depending on the
polarisation of the outgoing $\rho$-meson, both at leading twist. Our analysis
covers both neutral and charged $ \rho $-mesons. We find that the order of
magnitude of the obtained cross sections are sufficiently large for a dedicated
experimental analysis to be performed, especially at JLab, for both
longitudinally and transversely polarised $\rho$. Furthermore, we compute the
linear photon beam polarisation asymmetry {which is} sizeable for a
longitudinally polarised meson. These predictions are obtained for both
asymptotic distribution amplitude (DA) and the \textit{holographic} DA.
