You are referring to Beer's law, which describes radiation transmission at a single wavelength. However consider its meaning for the broad absorption band of CO2, whose graph is the shape of a wide lump with low shoulders. For parts of the band above some threshold, exponential absorption will have eliminated almost all the radiation, and further increase in concentration will not change this: you cannot go below zero. Parts of the band below that threshold will absorb very little radiation, and increasing the concentration will change this very little: double almost zero is still almost zero. Concentration change only makes a practical difference at the edges of the band, and the CO2 band's shoulders are so steep that even drastic concentration changes only make for small growth in the width of the total absorption band.
P.S. MODTRAN assumes a nonmoving atmosphere, so its temperature predictions are not to be believed. The actual atmosphere experiences stronger vertical convection as ground temperature rises, which in turn produces secondary effects on vegetation, evaporation, and clouds that affects temperature. Predicting the net effect on temperature requires an accurate biological/meteorological simulation. And those are mighty thin on the ground.
P.P.S. Convection effects are why greenhouses actually get hot: the glass obstructs natural convection from cooling the ground. Calling radiation effects the "greenhouse effect" is scientific fraud.
Reference? I don't see how that can be the case at all. The basic physics of radiation balance have no logarithms, and the naive expectation would be that the amount of escaping radiation depend exponentially on the CO2 column density.