  This is the first of two papers on computing the self-force in a radiation
gauge for a particle moving in circular, equatorial orbit about a Kerr black
hole. In the EMRI (extreme-mass-ratio inspiral) framework, with mode-sum
renormalization, we compute the renormalized value of the quantity
$h_{\alpha\beta}u^\alpha u^\beta$, gauge-invariant under gauge transformations
generated by a helically symmetric gauge vector; and we find the related order
$\frak{m}$ correction to the particle's angular velocity at fixed renormalized
redshift (and to its redshift at fixed angular velocity). The radiative part of
the perturbed metric is constructed from the Hertz potential which is extracted
from the Weyl scalar by an algebraic inversion\cite{sf2}. We then write the
spin-weighted spheroidal harmonics as a sum over spin-weighted spherical
harmonics and use mode-sum renormalization to find the renormalization
coefficients by matching a series in $L=\ell+1/2$ to the large-$L$ behavior of
the expression for $H := \frac12 h_{\alpha\beta}u^\alpha u^\beta $. The
non-radiative parts of the perturbed metric associated with changes in mass and
angular momentum are calculated in the Kerr gauge.
