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The BICEP3 Polarimeter is a small aperture, refracting telescope, dedicated to the observation of the Cosmic Microwave Background (CMB) at 95GHz. It is designed to target degree angular scale polarization patterns, in particular the very-much-sought-after primordial B-mode signal, which is a unique signature of cosmic inflation. The polarized signal from the sky is reconstructed by differencing co-localized, orthogonally polarized superconducting Transition Edge Sensor (TES) bolometers. In this work, we present absolute measurements of the polarization response of the detectors for more than $\sim 800$ functioning detector pairs of the BICEP3 experiment, out of a total of $\sim 1000$. We use a specifically designed Rotating Polarized Source (RPS) to measure the polarization response at multiple source and telescope boresight rotation angles, to fully map the response over 360 degrees. We present here polarization properties extracted from on-site calibration data taken in January 2022. A similar calibration campaign was performed in 2018, but we found that our constraint was dominated by systematics on the level of $\sim0.5^\circ$. After a number of improvements to the calibration set-up, we are now able to report a significantly lower level of systematic contamination. In the future, such precise measurements will be used to constrain physics beyond the standard cosmological model, namely cosmic birefringence.