  We compute the leading order contribution to radiative losses in the case of
spinning binaries with aligned spins due to their spin-orbit interaction. The
orbital average along hyperboliclike orbits is taken through an appropriate
spin-orbit modification to the quasi-Keplerian parametrization for nonspinning
bodies, which maintains the same functional form, but with spin-dependent
orbital elements. We perform consistency checks with existing PN-based and
PM-based results. In the former case, we compare our expressions for both
radiated energy and angular momentum with those obtained in [JHEP \textbf{04},
154 (2022)] by applying the boundary-to-bound correspondence to known results
for ellipticlike orbits, finding agreement. The linear momentum loss is instead
newly computed here. In the latter case, we also find agreement with the
low-velocity limit of recent calculations of the total radiated energy, angular
momentum and linear momentum in the framework of an extension of the worldline
quantum field theory approach to the classical scattering of spinning bodies at
the leading post-Minkowskian order [Phys. Rev. Lett. \textbf{128}, no.1, 011101
(2022), Phys. Rev. D \textbf{106}, no.4, 044013 (2022)]. We get exact
expressions of the radiative losses in terms of the orbital elements, even if
they are at the leading post-Newtonian order, so that their expansion for large
values of the eccentricity parameter (or equivalently of the impact parameter)
provides higher-order terms in the corresponding post-Minkowskian expansion,
which can be useful for future crosschecks of other approaches.
