I am not clear why the coarse grained state is a 'completely mixed state'. If the MERA structure is regarded as a unitary quantum operation (where the disentanglers are unitary and the isometries can also be regarded as unitary by adding ancilla $|0\rangle$ states). Then the coarse grained state of a thermal boundary state is a mixed state with a diagonal density matrix, but not necessarily a completely mixed state (unitary operation does not change the spectrum of the boundary state).
Another related issue is the eternal black holes in AdS, which correspond to entangled thermal states, where the temperature is not infinite and the BH entropy is just given by the thermal entropy of a system with a finite temperature. Maybe the size of the eternal black holes are restricted by $S_{BH}$ so it corresponds to a subset of the DOF of the boundary states and the coarse grained system still has an infinite temperature.