I did RRTM runs for 10,000 ppm H2O/330 ppm CO2, and then again with no CO2. Downwelling LW radiation dropped by less than 5% from 353 to 336 w/m^2.
Increasing CO2 to 560 ppm increased the greenhouse effect by about half a percent, from 353 to 355 w/m^2. This roughly translates to 1.4C warming (280K * 0.005) with a doubling of CO2.
However, the numbers are exaggerated because I did cloud free runs. If clouds were include, the effect of CO2 would be smaller.