However, there is a fairly huge difference between moon's sun-close-distance and sun-far-distance.
I did not find data to compare here. So I will not dig into that detail further to find (->which turns out about realto be in the range of 1%; see Edit below for details on temperature margins.)
There are orbital season phases though, which are passed through almost four times in a planet-year. During those near-sun/far-sun phases a considerable difference in temperature can be expected. If this is strong enough to impact habitability is open
On average the planet receives 25% less sun energy because of the increased distance to speculation1.25 AU (moves it halfway between our Earth and Mars, for reference). However, the Moon's orbit makes almost no difference at that distance (influence about 1%).
Using this graph of sun's energy per distance, we can conclude that at 1.425 AU the incoming energy level dropped by about a quarter or 25%.
Given your assumption that the atmosphere will be sufficient to support life on the moon, we could assume that it is of similar composition as that of our earthEarth. Earth's average atmospheric temperature is about 15°C.
According to this graph of temperature ranges during the seasons the ranges for the Planet and Moon can be derived by lowering the values by 25% (Earth for reference):
Phase |
Earth |
Planet/Moon |
Winter |
7-16 |
5-12 |
Summer |
10-21 |
7-16 |
Average |
15 |
11 |