We consider $mathcal{N}=(2,2)$ AdS$_3$/CFT$_2$ dualities proposed in the large central charge limit ($ctoinfty$) by Eberhardt. Here we propose the associated D1-D5 systems to be orbifolds of the standard $mathcal{N}=(4,4)$ systems, thereby elevating the dualities to the finite-$c$ level on the boundary and to the quantum level in the bulk. In particular, we show that our brane systems yield low-energy sigma models whose subleading central charges match earlier predictions from bulk one-loop supergravity computations. In the case involving the Enriques surface, the finite-$c$ sigma model has a non-trivial elliptic genus which we use to microscopically explain both the Bekenstein-Hawking entropy and the subleading logarithmic correction to it for the associated macroscopic black brane.
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