= Solution
Use a convective layer $T(z)=T_s-\Gamma z$ capped at an isothermal minimum $T_t$ at height $z_t$. Let the initial <optical depth> be $\tau(z)=\tau_se^{-z/H_\tau}$ and $\tau_t=\tau_se^{-z_t/H_\tau}$. Within the lapse-rate layer, $dB_\lambda/d\tau=B'\Gamma H_\tau/\tau$; above it the derivative vanishes. The supplied <Schwarzschild equation> therefore gives
$$
I_0=\pi B_\lambda(T_t)+\pi B'\Gamma H_\tau G(\tau_t,\tau_s).
$$
Assume the additional <carbon dioxide> multiplies the relevant opacity by $a=1.1$, leaving temperatures, the lapse rate and $H_\tau$ initially fixed. All material surfaces then acquire $a$ times their previous <optical depth>. Hence
$$
\boxed{\Delta I_0=\pi B'\Gamma H_\tau\big[G(1.1\tau_t,1.1\tau_s)-G(\tau_t,\tau_s)\big].}
$$
For $\tau_t\ll1$ and $\tau_s\gg10$, the specified approximation gives
$$
\boxed{\Delta T_{\rm BT}=\frac{\Delta I_0}{\pi B'}=-\Gamma H_\tau\ln1.1\simeq-0.09531\Gamma H_\tau.}
$$
For illustration, $\Gamma=6.5\,\mathrm{K\,km^{-1}}$ and $H_\tau=7\,\mathrm{km}$ give $-4.34\,\mathrm K$; these are illustrative, not supplied data. The inequalities put the effective emitting level $\tau\sim1$ inside the decreasing-temperature layer, well above an opaque surface and below the thin isothermal cap. Increasing opacity raises that emitting level by $H_\tau\ln1.1$, so it is colder. In a spectral window or above a temperature inversion the approximation and possibly the sign change. Opacity proportional to mixing ratio is also an assumption, not exact for every absorption mechanism.
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