I love how interested you are in this and your desire to work this out from first principles. I say the equation is not appropriate for the same reason we're having this discussion on the residence time of CO2, particularly anthropogenic. It doesn't account for multiple processes and time scales. The IPCC uses the Bern Model discussed here:

http://euanmearns.com/the-half-life-of-co2-in-earths-atmosph...

Frankly I don't care about the 97% agreement. I'm sure it is just as high if not higher in terms of biologists agreeing that evolution is occurring. It's a talking point to get people with no scientific training on board. Unfortunately it has the opposite result in many cases since it's a weak argument from authority.

Excellent point about water vapor particularly when it precipitates and release energy during the phase change. I'm on mobile so not as good at finding things, here's some information from AR4 with sources, AR5 may go into more detail:

https://www.ipcc.ch/publications_and_data/ar4/wg1/en/ch9s9-5...

9.5.4.2.1 Detection of external influence on precipitation

Mitchell et al. (1987) argue that global mean precipitation changes should be controlled primarily by the energy budget of the troposphere where the latent heat of condensation is balanced by radiative cooling. Warming the troposphere enhances the cooling rate, thereby increasing precipitation, but this may be partly offset by a decrease in the efficiency of radiative cooling due to an increase in atmospheric CO2 (Allen and Ingram, 2002; Yang et al., 2003; Lambert et al., 2004; Sugi and Yoshimura, 2004). This suggests that global mean precipitation should respond more to changes in shortwave forcing than CO2 forcing, since shortwave forcings, such as volcanic aerosol, alter the temperature of the troposphere without affecting the efficiency of radiative cooling. This is consistent with a simulated decrease in precipitation following large volcanic eruptions (Robock and Liu, 1994; Broccoli et al., 2003), and may explain why anthropogenic influence has not been detected in measurements of global land mean precipitation (Ziegler et al., 2003; Gillett et al., 2004b), although Lambert et al. (2004) urge caution in applying the energy budget argument to land-only data. Greenhouse-gas induced increases in global precipitation may have also been offset by decreases due to anthropogenic aerosols (Ramanathan et al., 2001).

Several studies have demonstrated that simulated land mean precipitation in climate model integrations including both natural and anthropogenic forcings is significantly correlated with that observed (Allen and Ingram, 2002; Gillett et al., 2004b; Lambert et al., 2004), thereby detecting external influence in observations of precipitation (see Section 8.3.1.2 for an evaluation of model-simulated precipitation). Lambert et al. (2005) examine precipitation changes in simulations of nine MMD 20C3M models including anthropogenic and natural forcing (Figure 9.18a), and find that the responses to combined anthropogenic and natural forcing simulated by five of the nine models are detectable in observed land mean precipitation (Figure 9.18a). Lambert et al. (2004) detect the response to shortwave forcing, but not longwave forcing, in land mean precipitation using HadCM3, and Gillett et al. (2004b) similarly detect the response to volcanic forcing using the PCM. Climate models appear to underestimate the variance of land mean precipitation compared to that observed (Gillett et al., 2004b; Lambert et al., 2004, 2005), but it is unclear whether this discrepancy results from an underestimated response to shortwave forcing (Gillett et al., 2004b), underestimated internal variability, errors in the observations, or a combination of these.