The eclipses of the moon by the giant planet should not have a major effect on the moon's climate. The body of my answer has five parts.
Part One: the Habitable Edge.
Apparently gas giant planets have an "habitable edge", and a moon orbiting closer to the planet than the habitable edge would have so much tidal heating from the planet that - combined with other sources of heat - the moon would overheat. A lot of liquid water would become water vapor, which is a greenhouse gas, and that would make the moon hooter and hotter and increase evaporation of liquid water, until all the water on the moon becomes vapor in the atmosphere.
Since by definition a habitable world is one with liquid surface water, an otherwise suitable moon that suffers such a runaway greenhouse effect will become uninhabitable.
I have read that the habitable edge of a gas giant planet would be at a distance of about 5 radii of the planet. So the shadow of the gas giant planet would be about two planetary radii, or one diameter, wide, where it falls on a moon's orbit at a distance of 5 radii from the center of the planet.
A moon orbiting at a distance of 5 planetary radii would have an orbit with a total length of 2 pi r, where r is 5 planetary radii. So the planetary orbit would 31.4159 planetary radii long. Thus the side of the moon facing the planet would spend half the orbit, or 15.70795 planetary radii, in sunlight, minus 2 planetary radii, or a total of 13.70795 planetary radii in Sunlight.
Part Two: Light Reflected from the Planet.
When the moon was between the star and the planet, the side of moon facing the planet would be in darkness, except for light reflected from the planet. Of course the moon would cast a shadow upon the planet, and the shadowed part of the planet wouldn't reflect light back on the moon. But the total side of the planet facing the star should be many times as large as the shadow cast by the moon, so that would make just a tiny reduction in the reflected sunlight received by the moon which would help heat up the moon.
Part Three: Move the Planet and Moon a Little Closer to the Star.
So the shadow of the planet upon the moon would have only a minor effect on the total amount of sunlight which the moon received. That can be compensated by moving the planet and the moon a little closer to their star so that the light of the star is a little more intense at that closer distance.
Part Four: Make the Moon orbit beyond the Habitable Edge.
And that is in the case of a moon orbiting at a distance of 5 planetary radii, right at the habitable edge for the planet. A moon orbiting right at the habitable edge of the planet would be right on the brink of suffering a runaway greenhouse effect, and so being in the planet's shadow for a small period of time during every orbit would help to keep the moon from suffering a runaway greenhouse effect.
And of course moons orbiting farther from their planets than the habitable edge would be in the shadows of their planets for smaller percentages of their orbits and would be cooled less by those periods in shadow.
Part Five" Orbital Planes and Axial Tilts.
Now consider orbital planes. One orbital plane would be the plane the planet orbits around its star in. Another orbital plane would the plane the moon orbits its planet in.
Gravitational effects between a gas giant planet and its moon should make the moon orbit in the equatorial plane of the planet.
If a planet's axis of rotation is at a right angle, 90 degrees, to the plane it orbits in, the planet's equatorial plane will be in the plane of it's orbit around the star. In that case the planet is said to have an axial tilt of zero degrees. If a planet's axis of rotation is in the plane of its orbit around its star, it will have an axial tilt of 90 degrees and the planet's equatorial plane will be at 90 degrees to the plane of the planet's orbit.
Because the Earth's axis of rotation has an axial tilt of about 23 degrees, and because the Moon's orbit is not the plane of Earth's equator, the moon is not eclipsed by the Earth every time it is on the far side of Earth from the Sun.
The moon is on the far side of Earth about 12 or 13 times in a year, but only has an average of two lunar eclipses each year.
It would be easy to design a star system where a moon wouldn't ever be eclipsed by its planet.
Draw a circle representing the gas giant planet. Make it two planetary radii wide. Draw parallel lines from the planet two radii apart to represent the shadow of the planet in space. And then measure five planetary radii from the planet.
Put a tiny moon right above the shadow of the planet, and at a distance of 5 planetary radii from the center the planet and then measure the angle necessary which should only be about 10 degrees or something. If a moon orbits in the equatorial plane of the planet and at the habitable edge of its planet, that is the number of degrees the planet's axial tilt needs to be for the moon to never be in the shadow of the planet. The farther the moon orbits beyond the habitable edge, the lesser the axial tilt of the planet needs to be.
The four giant planets in our solar system have varying axial tilts. Jupiter 3.13 degrees, Saturn 26.73 degrees, Uranus 97.77 degrees, Neptune 28.32 degrees. So it seems fairly probable that a gas giant planet with a habitable moon will have an axial tilt high enough for the moon to never be in the the planet's shadow.
Part Six: Atmospheric Distribution of Heat.
Planets orbiting in the habitable zones of red dwarf stars would probably be tidally locked to the stars, so that one side always faced the star and the other side always faced away from the star. And it has been feared that such worlds would loose all their atmospheres which would freeze solid on the eternally dark and cold sides of the planets.
n contrast to the previously bleak picture for life, 1997 studies by Robert Haberle and Manoj Joshi of NASA's Ames Research Center in California have shown that a planet's atmosphere (assuming it included greenhouse gases CO2 and H2O) need only be 100 millibar, or 10% of Earth's atmosphere, for the star's heat to be effectively carried to the night side, a figure well within the bounds of photosynthesis.[23] Research two years later by Martin Heath of Greenwich Community College has shown that seawater, too, could effectively circulate without freezing solid if the ocean basins were deep enough to allow free flow beneath the night side's ice cap. Additionally, a 2010 study concluded that Earth-like water worlds tidally locked to their stars would still have temperatures above 240 K (−33 °C) on the night side.[24] Climate models constructed in 2013 indicate that cloud formation on tidally locked planets would minimize the temperature difference between the day and the night side, greatly improving habitability prospects for red dwarf planets.[4] Further research, including a consideration of the amount of photosynthetically active radiation, has suggested that tidally locked planets in red dwarf systems might at least be habitable for higher plants.[25]
https://en.wikipedia.org/wiki/Habitability_of_red_dwarf_systems#Tidal_effects
Sop atmospheric distribution of heat should keep the dark sides of habitable moons from becoming too cold and the light sides from becoming too hot, especially since their day and night cycles would be a lot shorter than the eternal night and eternal day of tidally locked worlds.
Conclusion:
Although the climate of another world is hard to calculate the factors mentioned above should make it reasonable for a planetary mass moon orbiting a giant planet to not have significant cooling problems resulting from eclipses by the planet.
