  In a recent work, [Phys. Rev. D. 94, 104010 (2016)], hereafter Paper I, we
have numerically studied different prescriptions for the dynamics of a spinning
particle in circular motion around a Schwarzschild black hole. In the present
work, we continue this line of investigation to the rotating Kerr black hole.
We consider the Mathisson-Papapetrou formalism under three different
spin-supplementary-conditions (SSC), the Tulczyjew SSC, the Pirani SSC and the
Ohashi-Kyrian-Semerak SSC, and analyze the different circular dynamics in terms
of the ISCO shifts and the frequency parameter ${x \equiv (M \Omega)^{2/3}}$,
where $\Omega$ is the orbital frequency and $M$ is the Kerr black hole mass.
Then, we solve numerically the inhomogeneous $(2+1)D$ Teukolsky equation to
contrast the asymptotic gravitational wave fluxes for the three cases. Our
central observation made in Paper I for the Schwarzschild limit is found to
hold true for the Kerr background: the three SSCs reduce to the same circular
dynamics and the same radiation fluxes for small frequency parameters but
differences arise as $x$ grows close to the ISCO. For a positive Kerr parameter
$a=0.9$ the energy fluxes mutually agree with each other within a $0.2\%$
uncertainty up to $x<0.14$, while for $a=-0.9$ this level of agreement is
preserved up to $x<0.1$. For large frequencies ($x \gtrsim 0.1$), however, the
spin coupling of the Kerr black hole and the spinning body results in
significant differences of the circular orbit parameters and the fluxes,
especially for the $a=-0.9$ case. Instead, in the study of ISCO the negative
Kerr parameter $a=-0.9$ results in less discrepancies in comparison with the
positive Kerr parameter $a=0.9$.
