I'm trying to come up with a plausible atmosphere of alien world (Callisto moon) inhabited by insectoid race (they live in caverns underneath/inside the ice crust).
Do you mean a moon like Callisto or the actual moon Callisto of Jupiter?
Callisto has a very tenuous atmosphere composed of carbon dioxide. It was detected by the Galileo Near Infrared Mapping Spectrometer (NIMS) from its absorption feature near the wavelength 4.2 micrometers. The surface pressure is estimated to be 7.5 picobar (0.75 μPa) and particle density 4 × 108 cm−3. Because such a thin atmosphere would be lost in only about 4 years (see atmospheric escape), it must be constantly replenished, possibly by slow sublimation of carbon dioxide ice from Callisto's icy crust, which would be compatible with the sublimation–degradation hypothesis for the formation of the surface knobs.
A picobar is one trillionth of a bar. Since a bar is 0.986923 standard atmospheres of Earth, 7.5 picobar is approximately 7.4 trillionths of the atmospheric pressure of Earth.
Thus Callisto's atmosphere contains carbon dioxide in the amount of a few trillionths the density of Earth's atmosphere and no detectable amounts of other gases including Chlorine.
So even if the insects of Callisto live in caverns in the ice, chlorine gas in the caverns should slowly escape out of the caverns and build up densities many times that of carbon dioxide and great enough to have already been detected by Earth astronomers.
So possibly your story should be set hundreds of millions of years in the past or future of Callisto, when it has been terraformed by some advanced civilization to have an atmosphere containing chlorine gas dense enough to be breathed by the insects of Callisto.
And if your story is not set on Callisto, the moon of Jupiter, but a world like Callisto in some ways, there are a different set of problems to consider.
If your Callistan insects are intelligent beings they could have originated in another solar system and have some colonies and bases under the ice on Callisto. They could have excavated huge caverns and/or huge systems of smaller caverns connected by tunnels deep under the surface of the ice for the same reasons that human Moon bases are likely to be underground on the Moon.
They could obtain chlorine gas from salt in the subsurface ocean of Callisto. And their caverns and tunnels could be lined with gas tight materials to prevent the chlorine from seeping out and to the surface.
And they would use airtight vehicles to travel on the surface of Callisto and wear spacesuits for surface activities.
Of course if your Callistan insects are not intelligent beings but animals, they would have to live in natural caverns in the ice and I don't know where their Chlorine gas would come from.
Part Two: A World like Callisto.
What if your Callisto moon is not Calisto the moon of Jupiter, but a Callisto-like exomoon orbiting a giant exoplanet in another star system?
In that case your Callisto-like moon would be able to be unlike Callisto in some ways. But it would have to be more like Callisto that it was like other large moons in our solar system like the Moon of earth, Io, Europa, and Ganymede of Jupiter, and Titan of Saturn.
Of course the first think about Europa, Ganymede, Callisto, and Titan is that the rocks they are made of are largely water rocks, super cold and hard ice. Each of those worlds is believed to have a rocky core, surrounded by vast amounts of water, in layers of various exotic forms of ice, and quite probably an subsurface world wide ocean of liquid water supporting the uppermost ice layer.
And that means that none of those worlds has surface temperatures warm enough for liquid water. If they did their uppermost ice layers would melt and they would have world wide surface oceans instead of world wide icecaps tens or hundreds of kilometers deep.
So any one of those worlds could conceivably have lifeforms using liquid water in their subsurface oceans. But any hypothetical surface dwelling lifeforms would have to use chemical which are liquid at much lower temperatures than water for their water equivalents. So they would have to use liquid ammonia or liquid methane as their water equivalents depending on the surface temperatures.
You describe the Callistan lifeforms as "insects", but insects are a subcategory of Earth life, and you gave no indication that the Callistan "insects" are descended from insects from Earth. Thus they can't possibly be accurately described as insects. They are merely insect-like in appearance and/or structure, which means, for example, that you can give them lungs if you desire.
Since you describe the Callistan lifeforms as insects, it seems probable that they are surface dwelling lifeforms who burrow deep into caverns under the surface of the ice instead of squid like lifeforms from the interior oceans who have worked their way up from the oceans into caverns in the ice.
Thus they are not alien lifeforms with an exotic water drinking chlorine breathing biochemistry, but alien lifeforms living in much colder temperatures with an even more exotic ammonia or methane drinking chlorine breathing biochemistry.
The problem is how to give your Callisto-like world a dense enough atmosphere to breath.
How long a world can retain a gas in its atmosphere is primarily dependent on the escape velocity (and not, repeat not, the surface gravity) of the world and the temperatures in the exosphere of the atmosphere which determine the speed of gas particles in the exosphere.
In Habitable Planets for Man by Stephen H. Dole, 1964:
Dole discusses the retention of atmospheres by planets on pages 33 to 39. On pages 34 & 35 Dole discusses a rule of thumb for how long a planet can retain a gas in its atmosphere. The ratio of the escape velocity of the world divided by the root-mean-square of the velocity of that gas in the exosphere where gases escape into space indicates how long it would take for the amount of a gas to drop to 1/e o r0.368 of the original amount.
Table 5 on page 35shows that if the ratio is one or two, gas escape will be almost instant, while if it is three gas reduction to 0.368 will take a few weeks, if it is four gas reduction to 0.368 will take a few thousand years, if it is four gas reduction to 0.368 will take about a hundred million years, and if it is six gas reduction to 0.368 will take an infinite time.
Obviously if it takes a world a hundred million years to reduce the amount of a gas in the atmosphere to only 0.368 o the original amount, processes to replace that gas as fast or faster than it is lost into space are much more probable than if it takes only a few thousand years to reduce a gas to 0.368 of the original amount.
Of course Dole's interest is in how long a world will retain a human breathable atmosphere of oxygen.
Ganymede has an escape velocity of 2.741 kilometers per second, Callisto has an escape velocity of 2.441 kilometers per second, and Titan has an escape velocity of 2.641 kilometers per second.
A world described as Callisto-like might have an escape velocity slightly higher than Callisto's, but smaller than those of Titan and Ganymede.
With an escape velocity of 2.441 kilometers per second, it would be necessary for chlorine in the exosphere of your world's atmosphere to have a root-mean-square velocity of 0.4882 kilometers per second to have as much as 0.368 remaining after about a hundred millions years. It would be necessary for the chlorine to have a velocity of0.4068 kilometers per second or less to be retained for an infinite time.
If your Callisto like world had an escape velocity as high as 2.5 kilometers per second the velocities of Chlorine in the exosphere could be as high as 0.5 and 0.41666 kilometers per second.
One problem in modern planetary science is explaining how and why Titan has an atmosphere even thicker than Earth's while Ganymede and callisto, with similar escape velocities, have atmospheres a billionth as dense as Earth's.
The persistence of a dense atmosphere on Titan has been enigmatic as the atmospheres of the structurally similar satellites of Jupiter, Ganymede and Callisto, are negligible. Although the disparity is still poorly understood, data from recent missions have provided basic constraints on the evolution of Titan's atmosphere.
The exosphere temperature of Earth is several times as high as the air temperature at the surface. The temperatures in the exospheres of of planets and moons are believed to be cause by the ultraviolet radiation of their stars.
Jupiter, Ganymede, and Callisto orbit the Sun with a semi-major axis of 5.2038 AU, while Saturn and Titan orbit the Sun with a semi-major axis of 9.5826 AU. Orbiting at 1.841462 times the distance, Titan receives 0.2948997 as much ultraviolet radiation from the Sun as Ganymede and Callisto do. And possibly that is an important factor in the many times denser atmosphere of Titan.
I note that the temperature of a star determines which proportion of its radiation is emitted at specific wavelengths. The hotter the star, the higher the proportion of ultraviolet, and the cooler the star, the lower the proportion of ultraviolet.
So if your Callisto like moon orbited a planet in another star system, that star might be cooler than the Sun, and so emit a smaller fraction of its radiation in the ultraviolet. Thus a Callisto-like world might receive more total radiation than Titan and thus be somewhat warmer than Titan, though still very cold, while receiving no more ultraviolet than Titan does and so be as able to retain atmosphere as titan.
I note that the atmosphere of Titan is mostly nitrogen. The atomic weight of chlorine is more than twice that of nitrogen, so at the same temperature chlorine should have a slower speed than nitrogen and be easier for a world to retain in its atmosphere.
Of course chlorine is relatively rare in the universe compared to nitrogen or oxygen. But there are bound to be some examples of worlds with significant chlorine in their atmospheres.
Of course a writer doesn't have to care about the scientific plausibility of a Callisto-like world having an atmosphere with a lot of chlorine, or even worry about Callisto the moon of Jupiter having enough chlorine for aliens to breath despite chlorine not having been detected by space probes.
Some science fiction writers are content to have low scores on the scale of science fiction hardness.
And finally a quote:
You and I have drifted to the worlds that reel about the red star Arcturus, and inhabited the bodies of the insect philosophers who crawl proudly over the fourth moon of Jupiter."
"Beyond the Wall of Sleep" H.P. Lovecraft.