  We continue our study of large charge limits of the defect CFT defined by the
half-BPS Wilson loop in planar $\mathcal{N}=4$ supersymmetric Yang-Mills
theory. In this paper, we compute $1/J$ corrections to the correlation function
of two heavy insertions of charge $J$ and two light insertions, in the double
scaling limit where the charge $J$ and the 't Hooft coupling $\lambda$ are sent
to infinity with the ratio $J/\sqrt{\lambda}$ fixed. Holographically, they
correspond to quantum fluctuations around a classical string solution with
large angular momentum, and can be computed by evaluating Green's functions on
the worldsheet. We derive a representation of the Green's functions in terms of
a sum over residues in the complexified Fourier space, and show that it gives
rise to the conformal block expansion in the heavy-light channel. This allows
us to extract the scaling dimensions and structure constants for an infinite
tower of non-protected dCFT operators. We also find a close connection between
our results and the semi-classical integrability of the string sigma model. The
series of poles of the Green's functions in Fourier space corresponds to points
on the spectral curve where the so-called quasi-momentum satisfies a
quantization condition, and both the scaling dimensions and the structure
constants in the heavy-light channel take simple forms when written in terms of
the spectral curve. These observations suggest extensions of the results by
Gromov, Schafer-Nameki and Vieira on the semiclassical energy of closed
strings, and in particular hint at the possibility of determining the structure
constants directly from the spectral curve.
