The planet is massive, so you'll easily get tidal locking of the moon, that is the rotation of the moon's axis will be synchronous to the orbit duration (month) around the planet. For earth's moon, this duration is about 28 days, for your moon it will be a multiple of that.
The backside of your moon, turned away from the planet, will not get any sunlight for about half the moon's orbit duration, or full sunlight, the other half of the month.
The other side (turned toward the planet) would have no sunlight when the back side has.. and for a large part of the month, a solar eclipse occurs: the sun is covered completely by the huge planet.
I'm not sure about the inhabitable zone as a function of distance, but tidal lock could always present a problem for habitability by humans.. perceived day and night become very long !
On average, this moon will be very cold. There is no region inhabitable, because the "moderate zone" (or meridian) will also suffer from the lengthy solar eclipses, when the moon is behind the planet.
In order to be inhabitable, this moon should have
- its own rotation and less solar eclipse: a considerable distance to its planet, allowing every part of the moon getting sunlight and warm up, and
- a large orbit inclination angle i.r.t. to the planet's orbit around the sun. In case of tidal lock, you may have some narrow, butterfly-shaped zone with relatively moderate temperatures.
Moon formation scenario
Q: "the gas giant moved closer to the star and into the habitable zone as its moons were still forming, during this transit it would have captured a few massive bodies "
Suppose the above conditions are met, letting the moon have about the same daily cycle as earth and its orbit have a large inclination angle, solar eclipses will be rare.
Goldilock approach vector
When the small planet that becomes a moon is attracted by the big planet, it will proceed its orbit, but the vector of gravity will add to its movement.. in order to yield (eventually) a circular orbit, it should pass the planet with some kind of "goldilock vector": too close will mean the small planet will get a very elongated, elliptical orbit, too far away will just result in orbit disturbance and instability, it will not "take over" the planet easily. Between these cases, there should be a vector that results - in the end - in proper orbit. I've no means to put formulas or investigate..
Goldilock distance to the sun
The usual "Goldilock" condition also counts: distance to the sun. I started this answer worrying about the weather (temperature). I can add: at these masses/scales, it would require this giant planet to occupy a much closer distance to the sun. I wonder if a Goldilock-orbit would be allowed for this giant mass planet ? You could make it a blue star, or a heavier star with higher temperature, yielding some more energy..