= Solution
There are two necessary qualifications to the printed formula. \b[Critical points satisfy $g'=0$, not $g'=1$.] At a point with $g'=1$, the local degree is one and the printed summand is zero. In addition, $\log\|g'(0)\|$ is finite only when $g'(0)\ne0$; constant maps and origin-critical maps need separate treatment.
These are genuine issues even for otherwise ordinary disc maps. The map $g(z)=(z-1/2)^2/4$ maps the <unit disc> into itself and has $g'(0)\ne0$. Its <derivative> never equals one in the disc, so the printed count is zero, but it has a <ramification point> at $1/2$ of <multiplicity> one. For $R>1/2$ the correct identity has the additional term $\log(2R)$, proving that the displayed identity with the literal printed count is false. Also $g(z)=z^2$ shows the undefined basepoint logarithm in an origin-critical example.
Prove the intended identity first for a nonconstant map with $g'(0)\ne0$. Let $H(z)=2|g'(z)|/(1-|g(z)|^2)$ be its <hyperbolic derivative density> and $u=\log H$. Away from <derivative> zeros, $\log|g'|$ is harmonic, and direct differentiation gives
$$
\Delta[-\log(1-|g|^2)]=\frac{4|g'|^2}{(1-|g|^2)^2}=H^2.
$$
At a zero $a$ of $g'$ of <multiplicity> $m_a$, factor $g'=(z-a)^{m_a}h$ with $h(a)\ne0$. Its logarithm contains $m_a\log|z-a|$, whose distributional <Laplacian> is $2\pi m_a\delta_a$. Thus
$$
\Delta u=H^2+2\pi\sum_{g'(a)=0}m_a\delta_a.
$$
The local map degree is $m_a+1$, so this is the required excess-degree weighting.
For a smooth function, differentiating its circular mean and applying the <divergence theorem> gives $\overline u'(r)=(2\pi r)^{-1}\int_{|z|<r}\Delta u\,dA$. Integrating in $r$ gives the logarithmic Green-Jensen formula. Factoring the isolated logarithmic singularities extends it to the present $u$:
$$
\frac1{2\pi}\int_0^{2\pi}u(Re^{i\theta})\,d\theta-u(0)
=\frac1{2\pi}\int_{|z|<R}\log\frac R{|z|}\,\Delta u\,dA.
$$
Substitution yields the <critical-point Jensen identity for the hyperbolic derivative>
$$
\boxed{T_{\mathbb D}(R)+N_{\rm crit}(R)=\frac1{2\pi}\int_0^{2\pi}\log H(Re^{i\theta})\,d\theta-\log H(0),}
$$
where $N_{\rm crit}=\sum_{|a|<R:g'(a)=0}m_a\log(R/|a|)$. This is the requested formula with the critical-point typo corrected. Initially choose a circle containing no <ramification point> on its boundary; continuity and integrability of logarithmic boundary singularities extend the result to the remaining radii.
By <Schwarz-Pick theorem>, $H(z)\le2/(1-|z|^2)$. Therefore
$$
T_{\mathbb D}(R)+N_{\rm crit}(R)\le\log\frac2{1-R^2}-\log H(0).
$$
The normalized <hyperbolic distance> is $\rho(R)=\log((1+R)/(1-R))$, and
$$
\log\frac2{1-R^2}=\rho(R)+\log2-2\log(1+R)\le\rho(R)+\log2.
$$
Consequently
$$
\boxed{T_{\mathbb D}(R)+N_{\rm crit}(R)\le\rho(R)+\log\frac2{H(0)}.}
$$
One can take $C_1=1$ and $C_2=\log(2/H(0))\ge0$. Since the corrected count is nonnegative in the noncritical-basepoint case, this also proves the growth bound for $T_{\mathbb D}$ with the literal printed, identically zero count, though not its false identity.
For completeness, if $g'$ vanishes to order $m\ge1$ at zero, put $c=\lim_{z\to0}H(z)/|z|^m>0$ and
$$
N_{\rm crit}^{\rm reg}(R)=m\log R+\sum_{0<|a|<R:g'(a)=0}m_a\log(R/|a|).
$$
Apply the same formula to $\log H-m\log|z|$, which is regular at zero. It gives
$$
\boxed{T_{\mathbb D}(R)+N_{\rm crit}^{\rm reg}(R)=\langle\log H\rangle_R-\log c\le\rho(R)+\log(2/c).}
$$
The origin contribution is a regularized Jensen term, not the undefined $\log(R/0)$. If one instead omits the origin term, the upper bound gains $-m\log R$; this is bounded for $R\ge1/2$, while the characteristic and off-origin count are bounded on any smaller compact disc. Thus a bound $C_1\rho+C_2$ still holds for that nonnegative-count convention after enlarging $C_2$. For a constant map $H=0$ and $T_{\mathbb D}=0$, so its characteristic growth bound is trivial; the logarithmic identity and a finite critical-divisor count are not defined. These qualifications resolve all maps allowed by the original opening sentence without asserting an undefined equality.
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