I'm currently heavily revising a story which is largely set on a habitable but not exactly hospitable moon of a gas giant orbiting on the outer edge (or just beyond) of the habitable zone around a Sun-like star. I should note that the moon is not naturally habitable - it was terraformed in the distant past, and its ecosystem is a very simple engineered one.

For plot purposes, I need the moon to support unaided human life but get very cold very quickly when it passes into the gas giant's shadow and ideally stay this way for a day or two by Earth's standards. I've tried to research what factors might allow this to happen - distance to primary, the primary's diameter, inclination, atmospheric density, size of the moon, ocean coverage, cloud layers and so on but I've not been able to find a consistent answer.

How can I design this moon so that it is habitable but will get cold enough quickly enough to pose a significant threat to human life every time it enters the primary's shadow?

There is supposed to be little in the way of inclination so the eclipses are fairly regular events.