  Breakthrough experiments have recently realized fractional Chern insulators
(FCIs) in moir\'e materials. However, all states observed are Abelian, the
possible existence of more exotic non-Abelian FCIs remains controversial both
experimentally and theoretically. Here, we investigate the competition between
charge density wave order, gapless composite fermion liquids and non-Abelian
Moore-Read states at half-filling of a moir\'e band. Although groundstate
(quasi-)degeneracies and spectral flow is not sufficient for distinguishing
between charge order and Moore-Read states, we find evidence using entanglement
spectroscopy that both these states of matter can be realized with realistic
Coulomb interactions. In a double twisted bilayer graphene model transitions
between these phases by tuning the coupling strength between the layers: at
weak coupling there is a composite Fermi liquid phase and at strong coupling a
low entanglement state signaling CDW order emerges. Remarkably, however, there
is compelling evidence for a non-Abelian Moore-Read FCI phase at intermediate
coupling.
