  We present the first comparison of waveforms evaluated using the
effective-one-body (EOB) approach and gravitational self-force (GSF) theory for
inspiralling black hole binaries with a non-spinning primary and a spinning
secondary. This paper belongs to a series of papers comparing the EOB model
TEOBResumS to GSF results, where the latter are used to benchmark the EOB
analytical choices in the large-mass-ratio regime. In this work, we explore the
performance of two gauge choices for the gyro-gravitomagnetic functions GS, GS*
entering the spin-orbit sector within the EOB dynamics. In particular, we
consider the usual gauge of TEOBResumS, where GS and GS* only depend on the
inverse radius and the radial momentum, and a different gauge where these
functions also depend on the azimuthal momentum. The latter choice allows us to
exploit as prefactor in GS* the complete expression GKS* for a spinning
particle on Kerr. As done previously, we employ both waveform alignments in the
time domain and a gauge-invariant frequency-domain analysis to gain a more
complete understanding of the impact of the new analytical choice. The
frequency-domain analysis is particularly useful in confirming that the
gyro-gravitomagnetic functions in the new chosen gauge bring the EOB spin
contribution at 1st post-adiabatic order closer to the GSF one. We finally
implement the improved functions within the public code for TEOBResumS-Dal\'i,
which already incorporates eccentricity. In this way, we upgrade the EOB model
for extreme-mass-ratio inspirals presented in our previous work.
