We present a feasibility study of the longitudinal double-spin asymmetry A_LL in inclusive J/psi production in polarized proton-proton collisions at sqrt(s) ~ 27 GeV at the Spin Physics Detector (SPD) of the Nuclotron-based Ion Collider fAcility (NICA). At these moderate energies, J/psi production is dominated by gluon-gluon fusion, probing gluon momentum fractions x ~ 0.1-0.2 at central rapidity and highly asymmetric configurations at forward rapidity, where one parton can reach x ~ 0.5-0.9. This provides direct sensitivity to the poorly constrained moderate- to large-x region of the gluon helicity distribution Delta g(x). We estimate A_LL as a function of transverse momentum and rapidity using polarized parton distribution functions, focusing on the underlying partonic spin asymmetry. Nonperturbative long-distance effects are treated in a simplified manner and largely cancel in the asymmetry, enabling a direct assessment of gluon polarization sensitivity. We find asymmetries reaching |A_LL| ~ 0.09 at p_T = 3 GeV, with enhanced sensitivity at forward rapidity. The dominant theoretical uncertainty arises from polarized parton distribution functions. These results demonstrate that inclusive J/psi measurements at SPD/NICA provide a sensitive and complementary probe of gluon polarization at moderate and large x, extending constraints from RHIC into a kinematic regime not directly accessible to the EIC.