I cannot speak to Dr. Frank's assessment, but many of the models use Hansen's conversion of energy imbalance to change in temperature.
The forcing temperature change is .3 C per Watt per meter squared of imbalance.
This is per the IPCC and ACS,
Climate Sensitivity - American Chemical Society
ΔT ≈ [0.3 K·(W·m–2)–1] (2.2 W·m–2) ≈ 0.7 K
Hansen claimed in a paper the conversion for post ECS was more like .75 C per Watt per meter square of imbalance.
All of the models assume a 2XCO2 imbalance of 3.71 Wm-2.
The most basic portion of the concept is that doubling the level of CO2 causes an energy imbalance and forces a temperature change.
but the amount of the actual forcing imbalance is itself an assumption.
Feldman, observed a change in downwelling longwave radiation over a change in CO2 level,
http://asl.umbc.edu/pub/chepplew/journals/nature14240_v519_Feldman_CO2.pdf
but the 2XCO2 number would only come out to 2.52 Wm-2.
I personally think the people who programed the test in the HITRAN database, were not familiar,
with how population inversions can limit how quickly CO2 can cycle back to ground state.
Also how 2325 cm-1 transfer from sunlight excited nitrogen could break the Hitran modeled process.
Since the feedbacks are supposed to be an amplification of the forcing warming, an error in the forcing
warming would be magnified into a systematic error.