It is possible that the correct design for a habitable Earth-sized exomoon orbiting a gas giant planet in the habitable zone of their star might help with making this possible.
You should look up previous questions and answers about possibly habitable exomoons of gas giant planets.
One thing which they say is that for the orbit of a moon around its planet to be stable long term the orbit of the planet around its star has to last at least nine times as long as the orbit of the moon around its planet. So if the orbit of the moon around its planet lasts 10 Earth days, the orbit of the planet around its star would have to be at least 90 Earth days long.
And if the planet's orbit around the star is not much more than nine times as long as the moon's orbit around the planet, the combination of them might keep parts of the moon in sunlight or in the shadow of the planet for longer periods of time and thus cause those parts of the moon to heat up or cool down more.
Remember that the earth sized moon will be tidally locked so that it will rotate with the same period as it orbits around the planet. Thus there will be a planetward side of the habitable moon that will always face the planet, and an antiplanetward side of the moon that will always face away from the planet.
The shape of the shadow of the planet will depend on the values of the diameter of the star, the diameter of the planet, and distance between the star and the planet.The umbra, the totally dark part of the shadow, should be a cone that gets thinner and thinner with distance from the planet and comes to a point, while the penumbra, the partially dark part of the shadow, should be a cone that spreads out with distance from the planet, and extends out into space forever.
When the moon is in the penumbra the reduced light from the star may be enough to make water in the colder parts of the moon, such as in a hypothetical high equatorial ridge around the moon, freeze. And possibly if the sizeof the umbra at the distance of the moon is just right, only the equatorial ridge regions will ever been in total darkness and get even colder whenever the moon enters the umbra.
You might try modelling two different designs of the star/planet/moon system to see which gives the hottest poles and the coldest equator.
In both models the moon will orbit around the equator of the gas giant planet and the moon's own equatorial plane will be in the same plane as it orbits the gas giant planet, because tidal interactions between the moon and the planet will alter the moon's orbit and rotation into that configuration billions of years before the moon has a breathable atmosphere or advanced multi celled lifeforms such a intelligent beings evolve on that moon.
In one model the orbit of the moon around the gas giant planet should be in the same plane as the orbit of the gas giant planet around the star.
In the other model the orbit of the moon around the gas giant planet should be in a very different plane - titled about 90 degrees, similar to the axial tilt of Uranus - from the orbit of the gas giant planet around the star.
My answer here Polar Heat, Equatorial Cold - Climatic Effects of Inverted Global Temperatures1 discusses how the second model would affect the climate and possibly result in cold equator and hot poles.
There have been many questions about possible Earth sized habitable moons of gas giant planets.
I have answered a number of those questions.
Here is a link to a previous question and my answer to that question includes a link to a question and answer which has links to previous questions and answers.
What are the day and night fluctuations for a moon orbiting a planet the size of Jupiter?2
The article "Exomoon Habitability Constrained by Illumination and Tidal heating" by Rene Heller and Roy Barnes Astrobiology, January 2013, discusses factors affecting the habitability of exomoons.
Also see my answer here:
How long will it take to discover they live on a moon and not on a planet?4