The black body radiation of earth is proportional to the temperature to the 4th power so the % increase in incoming radiation (and outgoing since earth is in equilibrium) is proportional to the 4th power of the increase in temperature. Or IOW, the the temperature increase is proportional to the fourth root of the radiation increase. I'm sure you are aware of this, so I am clueless as to why you are saying they are equal.
What makes you so certain that approximating Earth with blackbody radiation equations alone is accurate? Earth doesn't even come close to meeting the definition of a "perfect blackbody".
Using long term temperature and long term solar irradiance records are the only possible way to even approximate the deviation from "perfect", and the graph that I showed above, and other similar temp vs. irradiance records, suggest that using a linear relationship is much more accurate than blackbody, at least in the short times (a couple thousand years) that we have that data. Clearly there are processes on Earth-Sun that lack thermal equilibrium, and lags resulting from that invalidate the strict use of blackbody. Lags that range from minutes to days to years to decades to millenia. Of course there is no simple comparison that will match the solar data to the temperature data! Yet, for the times we have available there are some striking similarities in the behavior - now even INCLUDING the last couple decades now that it is clear that aerosols have likely been suppressing the temperature response to the Sun since 1940.
The computations you make above only point out the problems with using strict blackbody approximations for Earth. They don't prove in any way that the .2% of increased irradiance is likely to be wholly responsible for the temperature changes... or at least, for on the order of 85% of the change.