Part One of Seven.:
Damarian, the moon of Checuti, will probably be tidally locked to Checuti, with one side probably always facing Checuti and the other side always facing away from Checuti. This means that it will take, about one orbit of Damarian around Checuti for the star or stars to appear to make a full circle around Damarian.
A synodic day of Damarian is likely to equal the length of its orbit around Checuti, except that during one orbit of Damarian around Checuti the planet Checuti will move some distance around its star. Thus the directin to the star will change slightly during one orbit of Damarian around Checuti, and Damarian will have to orbit for a slighly longer periord for the angle to the star to be the same as before.
So the orbit of Damarian around Checuti has to be short enough that the days and nights of Damarian are not too long for habitabilty.
So you should try to figure out how long the days and nights of Darmarian can be before the heat during the day and the cold during the night are too much for life to exist there.
If you want Damarian to be habitable for humans from Earth, you should check Habitable Planets for Man, Stephen H. Dole, 1964,for what he says about the range of lengths of days on a habitable world.
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf[1]
Dole considers rotation rates for a planet on pages 58 to 61 and decides that habitable worlds will have day lengths between about 2 or 3 hours and 96 hours (4 Earth days).
Part Two: The Year Length of Checuti and the Day Length of Damarian.
I note that the Earth takes about 365.256 Earth days to orbit the Sun once, thus travelling about 0.98561 degress of arc in an Earth day.
Since you say that Damarian's year is 384 to 385 (Damarian or Earth?) days long, and thus travels about 0.93689 degrees of arc per day, the correction you need to make between the sidereal and synodic days of Damarian, should be slightly smaller than for Earth.
"Exommon Habitability Constrained by Illumination and Tidal Heating" by Rene Heller and Roy Barnes, discusses some of the many factors affecting the potential habitability of exomoons orbiting around exoplanets in other star ssytems.
https://faculty.washington.edu/rkb9/publications/hb13.pdf[2]
One factor which they mention is:
The longest possible length of a satellite’s day compatible with Hill stability has been shown to be about P)p/9,
P)p being the planet’s orbital period about the star (Kipping,
2009a).
Kipping, D.M. (2009a) Transit timing effects due to an exomoon.
Mon Not R Astron Soc 392:181–189.3
This means that if the length of a orbital period around the planet, and thus its day, is more than one nineth as long as the planet's orbit around the star, the moon's orbit will be unstable and the moon wiil crash into the planet or escape from the planet in a short time by astronomical standards.
If the year of Checuti, the planet that Damarian orbits, is about 384.25 Earth days long, one nineth of a year should be about 42.694444 Earth days. So the maximum possible length of the month/day of Damarian should be about 42.694444 Earth days.
If you want Damarian to be habitable for Humans, and if Dole's estimate of a maximum of 4 Earth days is correct, Damarian must orbit Checuti in much less than the approximatley 42 days calculated in the pervious paragraph.
Their abstract of the article says:
. In addition
to radiative heating, tidal heating can be very large on exomoons, possibly even large enough for sterilization.
We identify combinations of physical and orbital parameters for which radiative and tidal heating are strong
enough to trigger a runaway greenhouse. By analogy with the circumstellar habitable zone, these constraints
define a circumplanetary ‘‘habitable edge.’’ We apply our model to hypothetical moons around the recently
discovered exoplanet Kepler-22b and the giant planet candidate KOI211.01 and describe, for the first time, the
orbits of habitable exomoons. If either planet hosted a satellite at a distance greater than 10 planetary radii,
then this could indicate the presence of a habitable moon.
This seems to indicate that they calculate that the habitable edge around a giant planet would be at a distance of about 10 planetary radii, and that moons inside that radius would heated too much and suffer a runaway greenhosue effect and become uninhabitable.
Since Jupiter has an equatorial radius of 71,492 kilometers, a habitable moon of a Jupiter-sized planet should have to orbit at a distance of at least 714,920 kilometers, if Heller and Barnes are correct.
Jupiter's moon Europa orbits Jupiter at a distance of 671,034 kilometers and has an orbital period and day 3.5512 Earth days long.
Jupiter's moon Ganymede orbits Jupiter at a distance of 1,070,412 kilometers and has an orbital period and day 7.1546 Earth days long.
That gives a rough idea of the minimum possible day length of Damarion if it orbits a planet with the equitorial radius of Jupiter.
Part Three: The Possible Maximum Size of the Planet Checuti.
So what would the radius of the planet Checuti be?
The original question says:
Planet: Checuti is about 25% larger than Jupiter (Both mass and diameter is best, but at least diameter), and has a ring system about 1.35 times as large as Saturn's.
A ficitonal giant planet can have a mass much greater than 1.25 that of Jupiter, but I don't know if it can have a radius 1.25 times the radius of jupiter.
The most massive giant planets can only have a radius slightly greater than that of Jupiter.
The more massive a planet is, the more its gravity conpresses the material at its core, and the greater its average density becomes. If more matter is added, the radius of the planet will increase, but not proportional to the matter added, as the overall density of hte planet become greater.
Eventually, with a planet a little more massive and with a little greater diameter than Jupiter, a limit is reached, and the planet stops growing as more mass is added, and instead becomes denser and denser and mabye even shrinks a little in radius.
So giant planets with up to about 13 times the mass of Jupiter are possible, the lower mass limit for what are called Brown Dwarfs. But the most massive planets, and brown dwars, and the least massive stars, all have approximately the same diameter as Jupiter.
The one exception is gas giant planets which orbit very close to their stars and are very hot, with atmospheres swollen to several times the normal diameter by the intense heat. Obviously a moon of such a planet would have a temperature of several thousand degrees and be far too hot for water based life forms.
You would have to consult an expert to find out what the greatest possible diameter of a gas planet with temperatures in the habitable range would be. But I doubt that it would be as much as 1.25 times the diameter of Jupiter.
Part Four: Minimum Possible Orbit and Day of Damarian.
If the planet Cheuti has a ring system about 1.35 times as wide as the rings of saturn, it should have a radius of and a diameter of
The equartorial radius of Saturn is about 60,300 kilometers and the main ring system extends from about 7,000 kilometers above the surface to about 80,000 kilometers above the surface. The main ring system has an outer radius of about 140,300 kilometes, and a diameter of about 280,600 kilometers. So a ring system 1.35 times as large would be a radius of 189,405 kilometers and a diameter of 378,810 kilometers.
So the moon Damarian would have to orbit the planet Checuti at a distance of at least 189,405 kilometers to be outside of the Roche limit, becuase the rings would probably be made of moons which passed within the Roche limit.
Or if Checuti had the equatorial radius of Jupiter, 71,492 kilometers, and the ring system extended to (80,000 times 1.35) kilometers about the surface of Checuti, the outer radius of the ring system would be about 179,492 kilometers above the center of Checkuti.
If Damarian orbited Checuti just beyond the ring system, it would have a very short day.
Saturn's moon Aegaeon orbits Saturn at a distance of 167, 500 kilometers and has an orbital period and day 0.80812 Earth days long.
Saturn's moon Mimas orbits Saturn at a distance of kilometers and has an orbital period and day 0.94242 Earth days long.
Jupiter's moon Amalthea orbits Jupiter at a distance of 181,366 kilometers and has an orbital period and day 0.5012 Earth days long.
Jupiter's moon Thebe orbits Jupiter at a distance of 222,452 kilometers and has an orbital period and day 0.6778 Earth days long.
Part Five: Maximum Possible Length of Damarian's Day.
If you want Damarian to be habitable for Earth humans and beings with similar requirements, and if you consider Dole's maximum day length of 4 Earth days to be correct, you would want Damarian to orbit Checutie in four days or less.
Saturn's moons Dione, Helen, and Polydeuces orbits Saturn at a distance of 377,396 500 kilometers and have orbital periods and days 2.736 Earth days long.
Saturn's moon Rhea orbits Saturn at a distance of 527,108 kilometers and has an orbital period and day 4.518 Earth days long.
Jupiter's moon Europa orbits Jupiter at a distance of 671,034 kilometers and has an orbital period and day 3.5512 Earth days long.
Jupiter's moon Ganymede orbits Jupiter at a distance of 1,070,412 kilometers and has an orbital period and day 7.1546 Earth days long.
If Checuti is much more massive than Jupiter, but has the same radius, the Moon Damarian could obit it at a somewhat greater distance and still have days less than 4 Earth days long.
Part Six: The Magnetospheres of Checuti and Damarian.
Unless the moon Damarian orbits within the magnetosphere of the planet Checuti, it will not be shielded by that magnetosphere from the damaging effects of solar winds and cosmic rays.
In "Magnetic Shielding of Exomoons Beyond the Habitable Edge" Rene Heller and Joge Zuluaga calcualte how the magnetospheres of giant planets form and expand over time. Some giant planets may extend their magnetospheres beyond their habitable edges to zones where habitable moons are possible in relatively short periods of astronomical time,and thus protect those moons from outside radiation.
https://arxiv.org/abs/1309.0811[4]
Moons at distances between about 5 and 20 planetary radii from a giant planet can be habitable from an illumination and tidal heating point of view, but still the planetary magnetosphere would critically influence their habitability.
5 to 20 times the equitorial radis of Jupiter would be 357,460 to 1,429,840 kilometers.
Jupiter's moon Thebe orbits Jupiter at a distance of 222,242 kilometers and has an orbital period and day 0.6778 Earth days long.
Jupiter's moon Europa orbits Jupiter at a distance of 421,700 kilometers and has an orbital period and day 1.7691 Earth days long.
Jupiter's moon Ganymede orbits Jupiter at a distance of 1,070,412 kilometers and has an orbital period and day 7.1546 Earth days long.
However, Heller and Zuluaga's alculations assumed that the potentially habitable exomoons might have only a few times the mass of Mars, and thus probably wouldn't have their own man gnetospheres, and would need to be protected by the mgeneto spheres of their planets.
If a moon could generate its own magnetosphere for protection, it would not have to orbit within the magnetosphere of the planet for protection.
The original question says.
Damarian's diameter is 19,113 km, but its gravity is nearly the same as Earth's (It' largely made out of a material denser than water, magnetic, and freezes well above a temperature habitable for the average Earth-like life form).
So Demarian could be large enough to have a strong magnetic field of its own and not need to orbit within the magnetic field of the planet Checuti. Thus Demarian could orbit at a greater distance from Checuti and have much longer orbital period and thus day.
However, long days and nights and high temperatures in the day and low temperatures in the night are rather inconsistent with habitability.
And as a general rule the faster a world rotates, and thus the shorter its day, the stronger its magnetic field will be. So Demarian needs to rotate relatively rapidly in order to generate a strong enough magnetosphere to not need to orbit within the magnetosphere of Checuti. So Demarian needs to have a relatively short day and thus orbit Checuti relatively close.
Part Seven: The Density of Damarian.
Damarian's diameter is 19,113 km, but its gravity is nearly the same as Earth's (It' largely made out of a material denser than water, magnetic, and freezes well above a temperature habitable for the average Earth-like life form).
The diameter of 19,113 kilometers is 1.5 times the 12,742 kilometers diameter of Earth, and thus Demarian has about 3.375 times the volume of Earth. If Demarian has 3.375 times the volume of Earth, and is made of the same substances as Earth, it should have at least 3.375 times the mass of Earth. The greater mass of Demarian will compress it's material to a greater average density than Earth.
However, it said that the gravity of Demarian should be about that of Earth.
On page 53 of Habitable Planets for Man, Dole says that since the upper limit of tolerabe surface gravity for humans is about 1.5 g, a terrestial planet with a surface gravity of 1.5 g would be the maximum size of a planet habitable for humans.
From Figure 9, it will be seen that this corresponds to a planet that has a mass of 2.35 Earth masses, a radius of 1.25 Earth radii, and an escape velocity of 15.3 kilometers per second.
I don't think that a surface gravity at the upper limit of human endurance counts as about the same gravity as Earth, but Demarian is supposed to be much larger than such a planet.
Thus Demarian must have an average density much less than that of Earth.
So the Roche limit of Checuti for an object with the average density of Demarian should be much farther from Checuti than it would be for an object with Earth's density.
Thus you need to avoid making Demarian too close to Checuti.
