I won’t comment on the final accuracy, but I will note that this is an extremely roundabout path to your final answer, and some of the intermediate steps are…weird. Most notably, the speculation that every man, woman, and child on the planet might run a 1 kW appliance 24/7/365. This is 7e13 kWh or 70k TWh, about 3x current global energy use (not just electicity) before accounting for efficiency. The equation you cite for radiative forcing, specifically its ln(new/old) term is very non-linear, so you should get a much lower marginal effect from 70k TWh than from 1 kWh.
A simpler approach is to calculate the CO2 required for your 1 kWh AC, i.e.: 1kWh * 3600 kJ/kWh / 0.6 efficiency / 890 kJ/mol = 6.7 mol CO2. Current atmospheric CO2 is 75 Pmol. From that, I get radiative forcing of ln((7.4e16 + 6.7)/7.4e16)/ln(2)3.7 * 4pi*(6.4e6^2). Numpy won’t tell me what ln(74000000000000006.7/74000000000000000). It will tell me the forcing from 10 kWh is ~2.5W, or the same 0.25W/kWh you got. I guess ln is not that nonlinear in the 1+1e-16 to 1+1e-4 range, after all.
0.25W/kWh seems improbably high. 1 kWh is about 0.1 W running 24/365. At 60% efficiency, that’s burning 0.2W of natural gas and implies that the radiative forcing from CO2 is much greater than the energy to produce the CO2 in the first place. I get that the energy source for heating is different from the energy source for electricity, but it feels wrong, even without the 1000 year persistence. I don’t know where the radiative forcing equation came from nor its constraints, so I’m suspicious of its application in this context. There’s a lot of obscenely large numbers interacting with obscenely small numbers, and I don’t know enough to say whether those numbers are accurate enough for the results to be reasonable. Then there’s the question of converting the energy input to temperature change.