We developed a microscopic framework that connects relativistic nuclear density functional theory to high-energy reaction observables by incorporating many-body Bogoliubov wavefunctions, represented in Pfaffian form, into the Monte Carlo integration of the Glauber model. This approach enables a simultaneous treatment of nuclear pairing correlations and multiple-scattering effects beyond the optical limit approximation (OLA). The method is applied to reaction cross sections of even-even O, Ne, and Mg isotopes on a ${}^{12}$C target. Complete multiple-scattering calculation systematically reduces the cross sections relative to the OLA and improves the agreement with measured interaction cross sections. Applications to the deformed Ne and Mg isotopes show that calculations based on a single intrinsic orientation depend strongly on the angle between the nuclear symmetry axis and the beam direction. Angular-momentum projection removes the spurious dependence on the arbitrarily selected intrinsic orientation and produces smooth isotopic systematics for the laboratory-frame cross sections. This work establishes a microscopic bridge between relativistic nuclear structure theory and reaction dynamics, providing a general framework for exploring exotic phenomena such as halos and clustering.