That's physically impossible with CO2 and climate because the relationship is highly nonlinear and can only be (semiaccurately) modeled over a small range (current conditions measured by satellites with a small delta) with lousy results outside that range.
Nonetheless, the magnitude of the coupling needs to be known. Having modeled nonlinear systems myself I realize it's a much more difficult task than for linear systems. But it can be done. And you're right, you linearize within certain ranges were it has a somewhat linear characteristic.
And you must know where those limits and discontinuities are located.
According to the Vostok Ice Core Data of CO2/Temp vs Time, Temps had a significant lead over CO2 on the down slope. Something caused that. What? Could it have been caused by too high levels of CO2? The CO2 remained high for thousands of years while the Temps dropped. Why? Or was it the Sun?
If the coupling is insignificant or in the wrong direction, and/or the effects minuscule compared to other unknown naturally caused or for that matter unnaturally caused effects we really don't understand the processes with any reasonable degree of confidence. The degree of confidence that's required before we start recommending costly solutions.