= Solution
Let $q$ be the mass mixing ratio of the uniformly mixed infrared absorbers, and let $k$ be their constant mass absorption coefficient. Hydrostatic integration gives downward <optical depth> $\tau=kqp/g$. The useful fixed-tropopause-optical-depth approximation requires a sufficiently optically thick lower column, unchanged absorbed sunlight and upper radiative profile in optical-depth coordinates, and the same convective lapse law after the perturbation. Then the matching to the radiative region occurs at the same $\tau_t$ and temperature $T_t$. Doubling all those absorbers gives
$$
\boxed{p_t'\simeq p_t/2.}
$$
Assume also that the surface pressure is unchanged. A dry adiabat satisfies $T_t/T_s=(p_t/p_s)^\kappa$ with $\kappa=R_a/c_p=0.287$. Consequently
$$
\boxed{\frac{T_s'}{T_s}=2^{0.287}=1.2201,\qquad
\Delta T_s\simeq0.2201T_s\simeq63\,\mathrm K}
$$
for an initial Earth-like surface temperature $288\,\mathrm K$. Its first-order logarithmic estimate, $\Delta T_s\simeq\kappa T_s\ln2$, is about $57\,\mathrm K$; the finite-doubling formula is preferable because the perturbation is large. This estimate is conditional on fixed $\tau_t,T_t$ and dry convective adjustment, not on carbon dioxide's measured spectral behavior.
There is also a consistency limitation to the most literal opacity model. A self-contained plane-parallel grey atmosphere with constant mass opacity has $T^4\propto\tau+2/3$ and $\tau\propto p$. Its radiative logarithmic gradient is
$$
\nabla_{\rm rad}=\frac{d\ln T}{d\ln p}=\frac{\tau}{4(\tau+2/3)}<\frac14<\kappa.
$$
It is therefore subadiabatic for these dry heat capacities, so it does not by itself establish the assumed deep dry convective column. The preceding warming is the intended fixed-matching-level, convectively prescribed estimate. A fully self-consistent model needs opacity varying with pressure, other heating structure or another closure to supply that convection; the two statements should not be silently identified.
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