A Compton camera prototype for ion beam range monitoring via prompt (< 1 ns) gamma detection in hadron therapy is being developed and characterized at the Medical Physics Department of LMU Munich. The system consists of a large (50x50x30 mm3) monolithic LaBr3(Ce) scintillation crystal as absorber component to detect the multi-MeV Compton scattered photons, together with a stack of 6 double-sided silicon strip detectors (DSSSD) acting as scatterer component. Key ingredient of the γ-source reconstruction is the determination of the γ-ray interaction position in the scintillator, which is read out by a 256-fold segmented multi-anode photomultiplier tube (PMT). From simulations an angular resolution of about 1.5o for the photon source reconstruction can be expected for the energy range around 3 – 5 MeV, provided that a spatial resolution of 3 mm can be reached in the absorbing scintillator . Therefore, particular effort was dedicated to characterize this latter property as a function of the γ-ray energy. Intense, tightly collimated 137Cs and 60Co photon sources were used for 2D irradiation scans (step size 0.5 mm) as prerequisite for studying the performance of the “k-Nearest-Neighbors” algorithm developed at TU Delft  (together with its variant ”Categorical Average Pattern”, CAP) and extending its applicability into the energy range beyond the original 511 keV. In this paper we present our most recent interaction position analysis in the absorbing scintillator, leading to a considerably improved value for the spatial resolution: systematic studies were performed as a function of the k-NN parameters and the PMT segmentation. A trend of improving spatial resolution with increasing photon energy was confirmed, resulting in the realization of the presently optimum spatial resolution of 2.9(1) mm @1.3 MeV, thus reaching the design specifications of the Compton camera absorber. The specification goal was reached also for a reduced PMT segmentation of 8x8 anode segments (each with 6x6 mm2 active area), thus allowing to reduce the complexity of the signal processing while preserving the performance.