So do you mean a tidally locked planet? Because I couldn't find anything about a bulls-eye planet.
Assuming that you actually meant a tidally locked planet (no rotation around its own axis), that would be the following effects:
On the hot side, the air would expand since it's hotter. On the dark side, it would shrink since it's colder. This would result in high altitude winds from the hot side going to the dark side and in ground winds from the dark side going to the hot side. This would certainly balance the extreme temperature differences a bit, but I doubt that it would make it habitable everywhere.
The people living on such a planet would face some problems:
1. The wind: It would be a serious problem if winds would race over the countryside at Mach 1. I'm currently writing a story that's set on a tidally locked planet (let's call it TLP from now on) and there their ancestors that placed life on the planet also built high cliffs that went around the "belt" of the planet where there is a constant dawn. Those cliffs prevent the winds from instantly eradicating everything the people try to build down there in the habitable zone of the planet, where the temperatures go to neither extreme.
2. The temperature: On the bright side, the temperature is extremely hot. No one would survive this. Furthermore, on the center point of the hot side, the rising air would cause winds from every side that go up and most likely leave a weak vacuum on the ground.
The cold side is also too cold. Obviously. The habitable ring where there is something like a constant sunset (beautiful), the temperature would most likely be acceptable, depending on the distance from the sun and its size and temperature.
3. Water: If you have water, it would evaporate on the hot side and be carried along with the high altitude winds to the cold side where it would most likely freeze to small ice rubble which could be carried along with the ground winds to the habitable zone where it would start to rain, but only for a short distance, after which it would evaporate again. So the most likely water circle you're going to see on this planet is some kind of sea that freezes on one side and boils on the other and in the middle, we have a massive downpour that never stops.
To answer your idea with the eclipses: I highly doubt it. I mean, our moon doesn't cool the earth down with a serious effect. Of course, while the moon is blocking the sun if it's directly overhead, it's getting a bit colder (about 5°C) but as soon as the moon is gone, the temperature returns to normal in, like, half an hour. So no.
I'm not sure what percentage of the planet could be habitable, but I'd say that at any point where the sun is visible on the horizon to where it is almost fullly visible, the temperatures would be acceptable, maybe even higher (think about the poles of our planet). So this totally depends on the size of the sun and the distance of the planet.