We investigate and solve the weak noise theory for the semi-discrete O'Connell-Yor directed polymer. In the large deviation regime, the most probable evolution of the partition function obeys a classical non-linear system which is a non-standard discretisation of the nonlinear Schrodinger equation with mixed initial-final conditions. We show that this system is integrable and find its general solution through an inverse scattering method and a novel Fredholm determinant framework that we develop. This allows to obtain the large deviation rate function of the free energy of the polymer model from its conserved quantities and to study its convergence to the large deviations of the Kardar-Parisi-Zhang equation. Our model also degenerates to the classical Toda chain, which further substantiates the applicability of our Fredholm framework.