The possible habitability of exomoons orbiting exoplanets in other star systems is a topic in astrobiology, the theoretical study of the possibility of life on other worlds. If exomoons can be habitable, that would make a significant increase in the number of habitable worlds in the galaxy, and extend the mass range of stars capable of having habitable worlds.
Most stars in the galaxy are so dim that a planet orbiting in their habitable zone will be tidally locked to the star. One side will always face the star in eternal light and another side will always face away from the star in eternal darkness. Would such a planet be lifeless because of the heat on the sunward side and the cold on the dark side, where the atmosphere might freeze solid? Or would he air and water needed for a planet to be habitable transfer heat from the day side to the dark side, keeping the temperatures livable? That is a question.
But if a planet has a large enough companion world, the tidal interactions with the companion world should tidally lock the planet to the companion world, preventing the planet from becoming tidally locked to the star, and thus the planet would have alternating day and night.
In Habitable Planets for Man, 1964, Stephen H. Dole discussed the requirements for a planet to be habitable for humans, and thus for the natives of your world.
On pages 67 to 72 Dole discussed the necessary properties and mass range for stars capable of having human habitable planets. And on pages 72 to 75 Dole discussed how a planet in the habitable zone of a low mass star could become tidally locked to a massive companion world, preventing it from becoming tidally locked to its star and enabling it to have a succession of day and night.
And in the last couple of decades there has been a lot of discussion of the possible habitability of giant exomoons orbiting giant planet in the habitable zones of their stars.
So you should read a few scientific articles discussing the possibility of life on hypothetical giant exomoons of exoplanets.
For example, "Exomoon habitability constrained by illumination and Tidal heating" Rene Heller and Roy Barnes, Astrobiology, 2013. The introduction explains the reasons for considering the habitability of exomoons, for example.
Heller and Barnes introduced the concept of the habitable edge around a giant planet in that article. If a moon orbits its planet closer than the habitable edge, tidal heating of the moon will be excessive and the moon will lose its water from a runaway greenhouse effect.
You asked about the magnetosphere of your giant planet and its effects on the habitability of your giant moon.
Many planetary mass objects, or planemos, generate weaker or stronger magnetic fields by processes not yet completely understood, I think. And if a planemo has a magnetic field, and orbits close to a star, the magnetic field will interact with the charged particles in the solar wind to create a planetary magnetosphere.
Earth's magnetic field, predominantly dipolar at its surface, is distorted further out by the solar wind. This is a stream of charged particles leaving the Sun's corona and accelerating to a speed of 200 to 1000 kilometres per second. They carry with them a magnetic field, the interplanetary magnetic field (IMF).
The solar wind exerts a pressure, and if it could reach Earth's atmosphere it would erode it. However, it is kept away by the pressure of the Earth's magnetic field. The magnetopause, the area where the pressures balance, is the boundary of the magnetosphere. Despite its name, the magnetosphere is asymmetric, with the sunward side being about 10 Earth radii out but the other side stretching out in a magnetotail that extends beyond 200 Earth radii. Sunward of the magnetopause is the bow shock, the area where the solar wind slows abruptly.
Inside the magnetosphere is the plasmasphere, a donut-shaped region containing low-energy charged particles, or plasma. This region begins at a height of 60 km, extends up to 3 or 4 Earth radii, and includes the ionosphere. This region rotates with the Earth. There are also two concentric tire-shaped regions, called the Van Allen radiation belts, with high-energy ions (energies from 0.1 to 10 MeV). The inner belt is 1–2 Earth radii out while the outer belt is at 4–7 Earth radii. The plasmasphere and Van Allen belts have partial overlap, with the extent of overlap varying greatly with solar activity.
Thus the Earth's magnetosphere is many larger than the radiation belts which contain dangerous radiation. So a manned space station orbiting Earth, for example, can orbit relatively safely in most parts of the magnetosphere, as long as it orbits outside of the radiation belts.
And the magnetosphere of a giant planet like Jupiter should be similar in many ways to that of Earth.
The magnetosphere of Jupiter is the cavity created in the solar wind by the planet's magnetic field. Extending up to seven million kilometers in the Sun's direction and almost to the orbit of Saturn in the opposite direction, Jupiter's magnetosphere is the largest and most powerful of any planetary magnetosphere in the Solar System, and by volume the largest known continuous structure in the Solar System after the heliosphere. Wider and flatter than the Earth's magnetosphere, Jupiter's is stronger by an order of magnitude, while its magnetic moment is roughly 18,000 times larger. The existence of Jupiter's magnetic field was first inferred from observations of radio emissions at the end of the 1950s and was directly observed by the Pioneer 10 spacecraft in 1973.
The action of the magnetosphere traps and accelerates particles, producing intense belts of radiation similar to Earth's Van Allen belts, but thousands of times stronger. The interaction of energetic particles with the surfaces of Jupiter's largest moons markedly affects their chemical and physical properties. Those same particles also affect and are affected by the motions of the particles within Jupiter's tenuous planetary ring system. Radiation belts present a significant hazard for spacecraft and potentially to human space travelers
The four Galilean moons of Jupiter orbit with semi-major axes of 421,800 kilometers (Io), 671,100 kilometers (Europa), 1,070,400 kilometers (Ganymede), and 1,882,700 kilometers (Callisto).
Io and Europa are within a radiation belt, and a human on the surface of Europa would receive a lethal dose of radiation in a a few hours.
The radiation level at the surface of Ganymede is considerably lower than at Europa, being 50–80 mSv (5–8 rem) per day, an amount that would cause severe illness or death in human beings exposed for two months.
Callisto is a lot lower. It's out of the radiation belts, but still inside Jupiter's colossal magnetosphere, protected from galactic cosmic rays (another radiation source). A human on the surface of Callisto would receive even less radiation then in deep space and even less than on our Moon.
Its distance from Jupiter also means that the charged-particle flux from Jupiter's magnetosphere at its surface is relatively low—about 300 times lower than, for example, that at Europa. Hence, unlike the other Galilean moons, charged-particle irradiation has had a relatively minor effect on Callisto's surface. The radiation level at Callisto's surface is equivalent to a dose of about 0.01 rem (0.1 mSv) per day, which is over ten times higher than Earth's average background radiation.
So it is good for your moon to be within the magnetosphere of your planet, but bad to be within the radiation belts which are within the magnetosphere of your planet.
I don't know how to calculate what the best orbit would be for a moon orbiting a planet of a specific mass at a specific distance, the planet itself orbiting a specific star with a specific mass (and thus specific solar wind strength) at a specific distance.
Unless you can find a way to calculate that, you will just have to guess at a size and mass of your planet and its moon and the orbit of the moon that puts the moon inside the magnetosphere and outside the radiation belts.
Rene Heller and Jorge I. Zuluaga in "Magnetic Shielding of Exomoons beyond the circumplanetary habitable Edge" (2019) discusss the shielding of hypothetical Mars sized exomoons by the magnetospheres of their giant planets.
They claim that Mars sized exomoons would probably not have their own magnetospheres to protect them from stellar wind and cosmic radiation, and so would be dependent on the magnetosphere's of their planets for protection. So they discuss the formation and evolution of the magnetospheres of giant planets, and at what distances they might extend to cover orbiting moons before the stellar wind eliminates the atmospheres of the moons.
Of course a Mars size exomoon might be large enough to be habitable for some types of liquid water using life, but it would probably be too small to be habitable for humans anyway. There are many lifeforms on Earth which flourish where unprotected humans would swiftly die, so obviously moons or planets can be habitable for some liquid water using lifeforms without being habitable for humans.
so you would want to use a more rare Earth mass exomoon as the setting, as described in the question. Such a large moon could probably generate its own magnetic field and its own magnetosphere and thus could orbit outside of the magnetic field of the planet. But since science has not yet fully explained the strength or weakness of the magnetic fields of all he moons and planets in our solar system, we can't be certain, so you might want to put the planet inside the magnetosphere of the giant planet anyway.
A planet can only hold moons in stable orbits if they orbit at or less than about 0.333 to 0.50 of the radius of the planet's Hill sphere. The radius of a planet's Hill sphere is calculated from the mass of the planet, the mass of the star, the semi-major axis of the planet's orbit, and the eccentricity of the planet's orbit.
It is possible that some giant planets will have magnetospheres extending as far as 0.333 to 0.5 of their Hill radius and so all permanent moons would have to orbit within the magnetosphere anyway.
Since you moon will probably be tidally locked to its planet, it will have one day night cycle during one orbit of the planet. Thus the closer the moon orbits the planet, the shorter its diurnal cycle will be, and the father it orbits from the planet, the longer the diurnal cycle will be, possibly becoming too hot for life in the day and too cold for life during the night.
Anyway, these are my thoughts on the subject.