  The dynamics of the planar circular restricted three-body problem with
Kerr-like primaries in the context of a beyond-Newtonian approximation is
studied. The beyond-Newtonian potential is developed by using the
Fodor-Hoenselaers-Perj\'es procedure. An expansion in the Kerr potential is
performed and terms up-to the first non-Newtonian contribution of both the mass
and spin effects are included. With this potential, a model for a test particle
of infinitesimal mass orbiting in the equatorial plane of the two primaries is
examined. The introduction of a parameter, $\epsilon$, allows examination of
the system as it transitions from the Newtonian to the beyond-Newtonian regime.
The evolution and stability of the fixed points of the system as a function of
the parameter $\epsilon$ is also studied. The dynamics of the particle is
studied using the Poincar\'e map of section and the Maximal Lyapunov Exponent
as indicators of chaos. Intermediate values of $\epsilon$ seem to be the most
chaotic for the two cases of primary mass-ratios ($=0.001,0.5$) examined. The
amount of chaos in the system remains higher than the Newtonian system as well
as for the planar circular restricted three-body problem with
Schwarzschild-like primaries for all non-zero values of $\epsilon$.
