Be X-ray binaries (Be-XRBs) are one of the largest subclasses of high-mass X-ray binaries, comprised of a rapidly rotating Be star and neutron star companion in an eccentric orbit, intermittently accreting material from a decretion disk around the donor. Originating from binary stellar evolution, Be-XRBs are of significant interest to binary population synthesis (BPS) studies, encapsulating the physics of supernovae, common envelope, and mass transfer (MT). Using the state-of-the-art BPS code, POSYDON , which relies on precomputed grids of detailed binary stellar evolution models, we investigate the Galactic Be-XRB population. POSYDON incorporates stellar rotation self-consistently during MT phases, enabling detailed examination of the rotational distribution of Be stars in multiple phases of evolution. Our fiducial BPS and Be-XRB model aligns well with the orbital properties of Galactic Be-XRBs, emphasizing the role of rotational constraints. Our modeling reveals a rapidly rotating population ( ω / ω _crit ≳ 0.3) of Be-XRB-like systems with a strong peak at intermediate rotation rates ( ω / ω _crit ≃ 0.6) in close alignment with observations. All Be-XRBs undergo an MT phase before the first compact object forms, with over half experiencing a second MT phase from a stripped helium companion (Case BB). Computing rotationally limited MT efficiencies and applying them to our population, we derive a physically motivated MT efficiency distribution, finding that most Be-XRBs have undergone highly nonconservative MT ( β ¯ rot ≃ 0.15 ). Our study underscores the importance of detailed angular momentum modeling during MT in interpreting Be-XRB populations, emphasizing this population as a key probe for the stability and efficiency of MT in interacting binaries.