Why in the world WOULDN'T the temperature go down? You are removing an energy source from the system.
The idea that the IPCC "downplays" the role of solar radiation is complete, utter nonsense. Take away solar radiation and the planet freezes very rapidly.
EDIT: I don't know what you mean when you say that "the Greenhouse Effect and CO2 are supposed to over power the Sun's TSI". When figuring out what the temperature will be at the Earth's surface, you need to consider ALL the factors that contribute: the solar radiation, convection, upward blackbody radiation and downward blackbody radiation. The current forcing from the change in CO2 is about 2 watts per square meter, at the grand solar minimum, the forcing from the change in the solar radiation will be less than 0.5 watts per square meter, so clearly that isn't going to cause any large cooling of the planet.
On the other hand, if it's close to noon during a solar eclipse, you're doing to drop one of the energy sources at that point by HUNDREDS of watts per square meter, so sure it's going to cool down some.
I understand what you're saying when you say that the sun is the only energy source, but it's misleading you. Try this experiment: on a hot summer day, lay your arm on the hood of a car and see how long you can keep it there. I'm guessing not very long. But why? You say that the Sun is the only source of energy and the side of your arm that is on the hood of the car is facing AWAY from the sun. What's happening of course, is that energy is being conducted from the hot metal of the car to your skin. If you left it there long enough, you'd get burned. Ultimately, it was the sun that warmed up the hood of the car, but at that moment it's energy from the hot car that's burning your skin, not energy from the sun. Ultimately, the source of most of the energy in the system is from the sun, but there are energy fluxes that are both upward and downward.