We have studied an energy gap opening at the Kramers point of quasi-2D Rashba semiconductor BiTeI with magnetic doping and influence of circularly polarized synchrotron radiation (SR) on the induced out-of-plane spin polarization of the gapped state. By means of angle- and spinresolved photoemission spectroscopy we have shown that below a Curie temperature, at 15-20 K, a spontaneous anomalously large energy gap at the Kramers point appears up to 90 and 125 meV depending on the V concentration (0.5 and 2%, respectively). Nevertheless, spin-resolved measurements show only a weak out-of-plane spin polarization both for the V 3d-resonances and the Rashba states owing to the presence of magnetic domains with opposite magnetic moments spontaneously generated without external magnetic field. Above a Curie temperature the out-of-plane spin polarization for the V 3d-resonances and 2D Rashba electron gas can be also induced by circularly polarized SR reversed in dependence on the chirality of circular polarization. It is followed by opening the energy gap at the Kramers point that confirms the induced magnetization. We connect the SR-induced out-of-plane spin polarization with a SR-derived hole generation leading to corresponding uncompensated spin accumulation in 2D Rashba electron gas with transferring the induced torque to the diluted V 3d-ions. The theoretical estimations corroborate well the experimental results.