In this paper, we introduce a computational procedure that enables autonomous LEO laser trackers endowed with INSs to increase the current accuracy when shooting at middle distant medium-size LEO debris targets. The code is designed for the trackers to throw the targets into the Atmosphere by means of ablations. In case that the targets are eclipsed to the trackers by the Earth, the motions of the trackers and targets are modeled by equations that contain post-Newtonian terms accounting for the curvature of space. Otherwise, when the approaching targets become visible for the trackers, we additionally use more accurate equations, which allow to account for the local bending of the laser beams aimed at the targets. We observe that under certain circumstances the correct shooting configurations that allow to safely and efficiently shoot down the targets, differ from the current estimations by distances that may be larger than the size of many targets. In short, this procedure enables to estimate the optimal shooting instants for any middle distant medium-size LEO debris target.