= Solution
Use the usual spherical <circular-shell adiabatic contraction> approximation: initial dark-matter orbits are circular, slow accumulation keeps them circular, and dark shells do not cross. <Angular momentum> is conserved in a spherical potential, and the vanishing <radial action> stays zero as an <adiabatic invariant>. For a circular shell,
$$
L^2=GrM(r),\qquad r_iM_i(r_i)=rM_f(r).
$$
Here $r_i$ is the initial shell radius and $r$ its final radius. The final flat <circular speed> $V=2v_{\rm NFW,c,max}$ gives $M_f(r)=V^2r/G$.
For an initial shell in the inner <NFW profile>, write $A=2\pi\rho_1r_1$, so $M_i(r_i)\simeq Ar_i^2$. The invariant gives
$$
Ar_i^3=\frac{V^2}G r^2,
\qquad r_i=\left(\frac{V^2}{GA}\right)^{1/3}r^{2/3}.
$$
Conservation of dark shell mass yields
$$
M'_D(r)=Ar_i^2=A^{1/3}\left(\frac{V^2}G\right)^{2/3}r^{4/3},
$$
and differentiation gives
$$
\rho'_D(r)=\frac{A^{1/3}}{3\pi}\left(\frac{V^2}G\right)^{2/3}r^{-5/3}.
$$
To express the requested ratios, set $K=F(x_{\max})\simeq0.2162166$ and $B=8K\simeq1.72973$, so $V^2/(GA)=Br_1$. Since the original cusp has $\rho_D(r)\simeq A/(2\pi r)$,
$$
\boxed{f(r)\simeq\frac23\left(\frac{Br_1}r\right)^{2/3}
\simeq0.96064\left(\frac{r_1}r\right)^{2/3}.}
$$
The total density of the final spherical flat-rotation model is $\rho_{\rm tot}=V^2/(4\pi Gr^2)$. Its dark fraction is
$$
h(r)\equiv\frac{\rho'_D}{\rho_{\rm tot}}
\simeq\frac43\left(\frac r{Br_1}\right)^{1/3}.
$$
Subtracting dark matter from the total gives the added-matter density, $\rho_B=\rho_{\rm tot}(1-h)$, and thus
$$
\boxed{g(r)\simeq\frac{\tfrac43(r/(Br_1))^{1/3}}
{1-\tfrac43(r/(Br_1))^{1/3}}
\sim1.11074\left(\frac r{r_1}\right)^{1/3}.}
$$
These expressions describe the leading inner approximation, not an extension to radii where the denominator becomes negative or the initial shell is outside the NFW cusp. As $r/r_1\to0$, dark matter is enhanced relative to its original density while its fraction of the even steeper total density tends to zero. This is <inner NFW contraction to a flat rotation curve>.
\b[The orbital assumptions matter.] An initial density profile does not specify the <phase-space distribution function>. For eccentric orbits, slow evolution conserves their actions, but $rM(r)$ at an instantaneous radius is not generally their invariant. The above ratios are the intended circular-shell estimate; their numerical coefficients cannot be uniquely inferred from the density profile and slow accumulation alone.
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