= Solution
The left side is atmospheric heat storage per unit surface area. Incoming solar flux $Q$ is averaged over the spherical surface by the factor $1/4$; <albedo> $a$ reflects a fraction of it. The outgoing term is a reduced Stefan-Boltzmann flux, with effective emissivity $\gamma$ representing atmospheric greenhouse trapping. <Ice-albedo feedback> usually makes $a$ decrease as <temperature> rises through an <ice>-melting range; greater carbon-dioxide <pressure> increases infrared opacity and therefore decreases this effective $\gamma$. The simple estimates below hold <albedo> fixed.
<Radiative equilibrium> gives
$$
\boxed{\gamma_0=\frac{(1-a)Q}{4\sigma T_0^4}\simeq0.5978.}
$$
Differentiating $\gamma T^4=\text{constant}$ gives $\Delta\gamma\simeq-4\gamma_0\Delta T/T_0\simeq-0.02474$. A doubling from $36$ Pa changes <pressure> by $36$ Pa, hence
$$
\boxed{\gamma_1\simeq\frac{4\gamma_0\cdot3}{290\cdot36}=6.87\times10^{-4}\ {\rm Pa}^{-1}.}
$$
Reference <weathering> balance requires $A_LW_0=v_0$, so $\boxed{W_0=2.0\times10^{-3}\ {\rm kg\,m^{-2}\,yr^{-1}}}$.
For the requested approximate system, take $(T_0,p_0)$ as the reference radiative/<weathering> <equilibrium>, set $\theta=(T-T_0)/\Delta T_c$, $P=p/p_0$, and $\hat t=t/t_{\rm CO_2}$, where
$$
\boxed{t_{\rm CO_2}=\frac{M_{\rm CO_2}Ap_0}{M_agv_0},\quad
t_T=\frac{\rho_ac_pd}{4\sigma\gamma_0T_0^3},\quad
\epsilon=\frac{t_T}{t_{\rm CO_2}},\quad
\Gamma=\frac{\gamma_1p_0T_0}{4\gamma_0\Delta T_c}.}
$$
Linearize the radiative loss, retaining terms linear in $T-T_0$ and $p-p_0$. Its two terms are $4\sigma\gamma_0T_0^3(T-T_0)$ and $-\sigma\gamma_1T_0^4(p-p_0)$. The carbon balance scales exactly with $v=v_{\rm CO_2}/v_0$. Therefore
$$
\boxed{\epsilon\theta_{\hat t}=-\theta+\Gamma(P-1),\qquad P_{\hat t}=v-P^\mu e^\theta.}
$$
Numerically $\boxed{t_{\rm CO_2}\simeq7.41\times10^3\ {\rm yr},\quad t_T\simeq0.0958\ {\rm yr},\quad\epsilon\simeq1.29\times10^{-5},\quad\Gamma\simeq0.1795}$. The time conversion uses seconds for the radiative constants and years for the supplied carbon production.
Because $\epsilon\ll1$, <temperature> rapidly approaches $\theta=\Gamma(P-1)$, while carbon changes slowly. A sustained $v\simeq40$ has <equilibrium> determined by
$$
\mu\log P_*+\Gamma(P_*-1)=\log40.
$$
Its left side is <strictly increasing> on positive <pressure>, giving $\boxed{P_*\simeq16.83,\quad\Delta T_{\rm eventual}=\Delta T_c\Gamma(P_*-1)\simeq36.9\ {\rm K}}$ within this extrapolated toy model. Counting volcanic production as well gives $v=41$ and $37.2$ K instead. An inventory estimate for the rise is $t_{\rm CO_2}(P_*-1)/(v-1)\simeq3.0\times10^3$ years. <Linearization> near the final <equilibrium> gives the <exponential relaxation time>[e-folding time]
$$
\boxed{t_{\rm relax}=\frac{t_{\rm CO_2}}{v(\Gamma+\mu/P_*)}\simeq9.4\times10^2\ {\rm yr}.}
$$
Thus the carbon/<weathering> response is millennial, despite the short <radiative relaxation time in a planetary atmosphere>[thermal adjustment] time.
There are two numerical-reference inconsistencies in the printed setup. The requested expansion treats $(T_0,p_0)$ as its reference <equilibrium> although earlier labels pair $T_0$ with present-day $36$ Pa rather than $p_0=28$ Pa. Also, the calibrated $\Gamma=0.1795$ gives $13^{0.3}e^{13\Gamma}\simeq22.3$, not the stated hint. That hint instead corresponds to $\Gamma\simeq3/13$, obtained by calibrating a doubling of $28$ Pa. Such a convention gives a <pressure> around $13$–$14$ and a similar rough warming of several tens of kelvin. The displayed results above consistently use the stated $36$ Pa doubling calibration and explicitly declared reference-model approximation. A large extrapolation of a linearized radiative law is a property of this exercise, not a quantitative climate projection.
This is a <weathering feedback with fast thermal adjustment>.
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