The quantum hydrodynamic model is used to study the nonlinear propagation of small amplitude magnetosonic solitons and their chaotic motions in quantum plasma with degenerate inertialess spin-up electrons, spin-down electrons, and classical inertial ions. Spin effects are considered via spin pressure and macroscopic spin magnetization current, whereas the exchange effects are considered via adiabatic local density approximation. By applying the reductive perturbation method, the Korteweg-de Vries type equation is derived for small amplitude magnetosonic solitary waves. We present the numerical predictions about the conservative system's total energy in spin-polarized and usual electron-ion plasma and observed low energy in spin-polarized plasma. We also observe numerically that the soliton characteristics are significantly affected by different plasma parameters such as soliton phase velocity increases by increasing quantum statistics, magnetization energy, exchange effects, and spin polarization density ratio. Moreover, it is independent of the quantum diffraction effects. We have analyzed the dynamic system numerically and found that the magnetosonic solitary wave amplitude and width are getting larger as the quantum statistics and spin magnetization energy increase, whereas their amplitude and width decrease with increasing spin concentration. The wave width increases for high values of quantum statistic and exchange effects, while their amplitude remains constant. Most importantly, in the presence of external periodic perturbations, the periodic solitonic behavior is transformed to quasiperiodic and chaotic oscillations. It is found that a weakly chaotic system is transformed to heavy chaos by a small variation in plasma parameters of the perturbed spin magnetosonic solitary waves. The work presented is related to studying collective phenomena related to magnetosonic solitary waves, vital in dense astrophysical environments such as pulsar magnetosphere and neutron stars.