Dispersions of colloidal platelets in the nematic phase display strong wall anchoring, which competes with the reorientational motion of the director when the system is subjected to flow. We show that the mechanical response to large amplitude oscillatory strain and stress depends on the confinement of the system due to this competition. We elucidate the underlying structural response by deflecting a x-ray beam vertically along the vorticity direction of a Couette geometry, such that the structure can be probed throughout the gap with an unprecedented spatial resolution while recording in situ the mechanical response. We observe strong inhomogeneities in terms of the orientation of the nematic director, depending on the extent of the system's yield during an oscillation. At small strain amplitudes, we observe a small region where the director oscillates between wall anchoring and the Leslie angle, while in the bulk, the director tilts out of the flow-flow gradient plane. At large strain amplitudes, the oscillations of the director are symmetric, close to the wall, and propagate into the bulk. Here, a twinning is observed where the director rotates out-of-plane in two opposite directions. Using the sequence of physical process method to analyze the LAOStrain response for both the mechanical and structural response, we locate the yielding in a small time-window around flow reversal and identify that the bulk is the main contributor to the mechanical response. The structural response to LAOStress is much less pronounced even when the stress amplitude causes significant shear thinning.