No, we know how much heat CO2 traps based on direct measurement
Your faith is strong, but it's misguided and based on popular oversimplifications. Here's Richard Lindzen writing in a WSJ editorial:
There are, however, some things left unmentioned about the IPCC claims. For example, the observations are consistent with models only if emissions include arbitrary amounts of reflecting aerosols particles (arising, for example, from industrial sulfates) which are used to cancel much of the warming predicted by the models. The observations themselves, without such adjustments, are consistent with there being sufficiently little warming as to not constitute a problem worth worrying very much about.
In addition, the IPCC assumed that computer models accurately included any alternative sources of warming—most notably, the natural, unforced variability associated with phenomena like El Nino, the Pacific Decadal Oscillation, etc. Yet the relative absence of statistically significant warming for over a decade shows clearly that this assumption was wrong. Of course, none of this matters any longer to those replacing reason with assertions of authority.