What irradiance does the Moon receive during a total lunar eclipse?
The following is more a series of observations than a full answer, but I hope other people can fill in the details.

The apparent magnitude of the full moon is -13. The apparent magnitude of the moon during a lunar eclipse will differ from eclipse to eclipse (correlated to the Danjon scale value), but let's suppose it is 0 for a certain eclipse. (Note that this magnitude is different from the (pen)umbral magnitudes mentioned on the NASA website; these magnitudes can be computed and do not depend on the Earth's atmosphere.)

This means the full Moon is normally reflecting $100^{13/5}$ times more light than during this eclipse. I'm not sure if the albedo of the Moon is different for red light than it is for normal sunlight, but if not, that means the answer is $\frac{1361}{100^{13/5}} \frac{W}{m^2} \approx 8.6 \frac{\text{m}W}{m^2}$.