Well this guy's webpage seems to report accurate air pressure measurements of the Grand Canyon: 790 millibars at the rim and 930 millibars at the bottom (a difference of 140 mb). Other sources on the internet report the average temperature difference between the rim and bottom to be approx. 25 degrees Fahrenheit (approx. 14 degrees Celsius/Kelvin). Since all other environmental factors would be the same at that location (latitude, humidity, atmospheric composition), it would seem as if every extra 10 millibars of pressure translates into an extra degree Centigrade/Kelvin, at least at Earth orbit. I'm not sure if it's a linear relationship all the way up into space, though.... Mount Everest's summit has an average air pressure of about 330mb, and Cairo about 1000mb (they are both on the 27th parallel north), suggesting an average temperature difference of 67 degrees. They aren't quite that far apart, but it's hard to tell what other factors might be at play that could effect the result. So.... you said close enough was good enough, right? 
As for atmospheric composition, this Wikipedia entry defines the most common greehouse gases on Earth as water vapour (H20), carbon dioxide (CO2), methane (CH4), nitrous oxide (N2O), Ozone (O3), and CFCs (I don't think they occur in nature in any volume...). Anyway, the effect that each gas has on global warming (or in your case, final atmospheric temperature potential) has to do with radiative forcing. There's lots of charts and data on that page.... I'd just get a feel for what is more important relative to its atmospheric concentration on Earth, and then extrapolate.
Hope that at least gives you some ideas....

As for atmospheric composition, this Wikipedia entry defines the most common greehouse gases on Earth as water vapour (H20), carbon dioxide (CO2), methane (CH4), nitrous oxide (N2O), Ozone (O3), and CFCs (I don't think they occur in nature in any volume...). Anyway, the effect that each gas has on global warming (or in your case, final atmospheric temperature potential) has to do with radiative forcing. There's lots of charts and data on that page.... I'd just get a feel for what is more important relative to its atmospheric concentration on Earth, and then extrapolate.
Hope that at least gives you some ideas....