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The orbital dynamics of fast spinning neutron stars encountering a massive black hole (BH) with unbounded orbits are investigated using the quadratic-in-spin Mathisson–Papapetrou–Dixon (MPD) formulation. We consider the motion of the spinning neutron stars with astrophysically relevant speed in the gravity field of the BH. For such slow-speed scattering, the hyperbolic orbits followed by these neutron stars all have near the e = 1 eccentricity, and have distinct properties compared with those of e ≫ 1. We have found that, compared with geodesic motion, the spin–orbit and spin–spin coupling will lead to a variation of scattering angles at spatial infinity, and this variation is more prominent for slow-speed scattering than fast-speed scattering. Such a variation leads to an observable difference in pulse-arrival-time within a few hours of observation, and up to a few days or months for larger BH masses or longer spinning periods. Such a relativistic pulsar-BH system also emits a burst of gravitational waves (GWs) in the sensitivity band of Laser Interferometric Space Antenna , and for optimal settings, can be seen up to |$100\, {\rm Mpc}$| away. A radio follow up of such a GW burst with SKA or FAST will allow for measuring the orbital parameters with high accuracy and testing the predictions of general relativity. [ABSTRACT FROM AUTHOR]