  We study the transition of $\nu=1/3$ and $2/5$ fractional quantum Hall states
of the honeycomb Hofstadter model as we tune to a two-orbital moir\'e
superlattice Hamiltonian, motivated by the flat bands of twisted bilayer
graphene in a perpendicular magnetic field. In doing so, we address the extent
to which these states survive in moir\'e systems and analyze the nature of the
transition. Through the use of a Peierls substitution, we determine the
Landau-level splitting for the moir\'e Hamiltonian, and study the structure of
the Chern bands for a range of magnetic flux per plaquette. We identify
topological flat bands in the spectrum at low energies, with numerically
tractable lattice geometries that can support the fractional quantum Hall
effect. As we tune the model, we find that the orbital-polarized $\nu=1/3$ and
$2/5$ states corresponding to the honeycomb Hofstadter model survive up to
$\approx 30\%$ of typical moir\'e superlattice parameters, beyond which they
transition into an insulating phase. We present evidence for this through
density matrix renormalization group calculations on an infinite cylinder, by
verifying the charge pumping, spectral flow, entanglement scaling, and
conformal field theory edge counting. We conclude that fractional quantum Hall
states from the Hofstadter model can persist up to hopping amplitudes of the
same order as those typical for moir\'e superlattice Hamiltonians, which
implies generally that fractional states for moir\'e superstructures can be
discerned simply by analyzing the dominant terms in their effective
Hamiltonians.
