= Solution
Take a weak line in the radiative sense: the integrated slant <optical depth> is small wherever <absorption> materially contributes, not merely a small greenhouse-gas <mass mixing ratio>. Normalize every local <spectral line profile> so that $\int\kappa_\nu(p)\,d\nu=S$, with the <Spectral line strength>[line strength] independent of <pressure> and, as stipulated, <temperature>. Keep a finite Doppler or natural core, or a <Voigt profile>, so that a pressure-only zero-width core at the top does not invalidate a uniform optical-depth expansion. The line must also be narrow enough to use $B_{\nu_0}$ across it.
To first order,
$$
E(p,q)\simeq\frac q{g\mu_d}\int_0^p\int\kappa_\nu(p')\,d\nu\,dp'
=\frac{qSp}{g\mu_d}.
$$
Thus the detailed normalized <Spectral line profile>[line shape] drops out, even if its width varies with <pressure>. Substituting in the deficit in <1/c> gives
$$
\boxed{\sigma_{\rm SB}T_s^4\simeq F_*+
\frac{\pi qSp_s}{g\mu_d}\left[B_{\nu_0}(T_s)-
\int_0^1B_{\nu_0}(T_s r^\kappa)\,dr\right].}
$$
The neglected term is second order when the relevant squared optical depths are integrable and uniformly small. A very narrow saturated line need not be weak simply because $q\ll1$. More generally the displayed first-order integrated limit follows from the finite integrated <opacity> even if an idealized <Spectral line>[line centre] makes a pointwise expansion nonuniform; a finite physical core is the straightforward sufficient assumption.
For an explicitly perturbative <temperature> estimate, evaluate the bracket at $T_{\rm bare}$:
$$
\boxed{T_s-T_{\rm bare}\simeq\frac{\pi qSp_s}{4g\mu_d\sigma_{\rm SB}T_{\rm bare}^3}
\left[B_{\nu_0}(T_{\rm bare})-\int_0^1B_{\nu_0}(T_{\rm bare}r^\kappa)\,dr\right]>0.}
$$
At fixed $p_s,g,q,S$ and radiative conditions, increasing $\kappa=R_b/c_{p,b}$ makes every level $0<r<1$ cooler relative to the surface, enlarging the bracket and the warming. Increasing $c_{p,b}$ at fixed $R_b$ has the opposite effect. The limit $\kappa\to0$ is <Isothermal process>[isothermal] and gives zero line-induced deficit. A change of background <molecular mass> alone need not change this leading result: for <Ideal gas>[ideal gases] with the same active <degrees of freedom>, $R_b$ and $c_{p,b}$ scale together and their ratio is unchanged. Comparing only the height-coordinate lapse rate $g/c_{p,b}$ would miss that the absorber column is naturally a pressure-coordinate quantity.
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