How do we allocate partial CO2 doublings for Empirical Climate Sensitivity?
To get a rough estimate you should be using a logarithmic interpolation rather than a linear interpolation. The response to added $\text{CO}_2$ would be linear if the atmosphere had barely any $\text{CO}_2$ in it. Even 280 ppm does not qualify as "very little". "Very little" would mean a handful of parts per million, at most.

The issue is that the atmosphere is rather opaque at those frequencies where $\text{CO}_2$ absorbs and emits thermal infrared radiation. Instead of a linear response, the response in terms of radiative forcing is roughly logarithmic. This is why climatologists talk about the effect of a doubling the $\text{CO}_2$ levels. The radiative forcing of an increase from 140 ppm $\text{CO}_2$ to 280 ppm is more or less the same as the response to an increase from 280 ppm to 560 ppm, or to an increase from 560 ppm to 1120 ppm.